View Full Version : Solitons in One Post [Was: Solitons in one sentence]
tessel@tum.bot
Jun13-04, 09:39 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Fri, 28 May 2004, Francois Belfort wrote:\n\n> An acquaintance told me that all one needs to know about solitons is\n> that they are a single travelling bumps,\n\nThe UW library catalog lists 72 books on solitons; in fact, there are\n-many- things concerning solitons which are well worth learning! I\'ll try\nto sketch a few of them in this post.\n\nThe particular equation you asked about is probably the most famous\nexample of a PDE exhibiting soliton solutions, the "Korteweg/de Vries\nequation", or "KdV equation" for short. This can be written\n\nu_t = u u_x + u_(xxx)\n\n(The form you gave is just an alternative "normalization"; you can easily\nfind a coordinate transformation taking that form into this one.)\n\nA fundamental point concerning the KdV equation is that it exhibits two\nopposing tendencies:\n\n1. "nonlinear convection", which tends to -steepen- wavecrests,\n\n2. "dispersion", which tends to -flatten- wave crests.\n\nThe idea is that by carefully balancing these two effects, it is possible\nto have a traveling wave, with a single wavecrest, which propagates\nwithout changing shape or losing energy. This is called a "solitary\nwave".\n\nTo understand this better, we should attempt to find the general traveling\nwave solution of the KdV equation. To do this, we plug the traveling wave\nAnsatz\n\nu(x,t) = f(x-ct)\n\ninto the KdV equation (more about this later) , obtaining the ODE\n\nf\'\'\'(w) = (c - f(w)) f\'(w)\n\nThis admits the integrating factor f(w), so we can reduce it to a second\norder autonomous (nonlinear) ODE:\n\nf\'\'(w) + f(w)^/2 - c f(w) + b = 0\n\nLet us study the phase portrait of this "once-reduced" ODE. We find that\nthere are two critical points, at\n\n{ (0,0) = saddle\n(f,f\') = {\n{ (2c,0) = center\n\nNote that in this phase portrait, we have a -sepatrix- which starts at the\norigin, winds once around the other critical point, and returns to the\norigin. Inside this sepatrix we have trajectories which are diffeomorphic\nto ellipses but "sharper" at their left hand ends. These correspond to\n-periodic solutions- of our ODE--- i.e. to periodic traveling wavetrains.\nThese periodic solutions may be written explicitly in terms of the Jacobi\nelliptic function cn(z; k) in the form\n\nu(x,t) = A cn(B (x-ct) + C, k)^2\n\nFor this reason, they were called "cnoidal waves" by Korteweg and de\nVries, who discovered them. As they noted, the period may expressed as an\nelliptic integral. Graphically, cnoidal waves look much like cosine\nwaves, but close examination shows they have sharper crests and flatter\ntroughs. Indeed, the function cn(.,.) also arises in solving the pendulum\nequation, whose linearization is the familiar harmonic oscillator\nequation, so their appearance here (and in many other places in the study\nof nonlinear DEs) is not too surprising. Using standard facts about\nJacobi elliptic functions, one can find an explicit "dispersion relation"\nbetween the wavenumber and speed of the periodic traveling waves---\nworking out the details is a good exercise in -nonlinear dispersion-.\n\nAs the period goes to infinity, the corresponding trajectories approach\nthe sepatrix from inside, which of course is a -non-periodic- solution.\nIt corresponds to the famous "one crested soliton" or "1-soliton" solution\nof the KdV equation:\n\nu(x,t) = 3 c sech^2 (sqrt(c)/2 (x-ct) + e)\n\nThis solution represents a disturbance with a single crest which travels\nindefinitely without changing shape, so it is a "solitary wave". But\nsaying it is a "soliton" implies much more! To see why, observe that from\nwhat we\'ve said so far, we might well expect that our solitary wave is\n-unstable-, i.e. that small perturbations should either change it into a\nlong period cnoidal wave, or into a disturbance which blows up.\nAlternatively, because this solitary wave arises from a governing equation\nwhich balances the tendency of nonlinear convection to steepen wave crests\nagainst the tendency of dispersion to flatten them (because any frequency\ncomponents will travel at different speeds, thus dispersing an initial\nwave profile), we might well expect solitary waves to be unstable---\nsurely this "balancing" is a delicate affair!\n\nBut in fact, decades of work (both experimental and theoretical, beginning\nwith Russell\'s work) show that solitons are stable against small\n-nonlinear- perturbations. This is, or should be, quite surprising!\n\nThe remarkable stability of solitons is well illustrated by their very\nsurprising nonlinear interactions. It turns out that we can find explicit\n"n-soliton" solutions to the KdV, with n crests of differing heights and\nspeeds; these solitary waves interact nonlinearly when they collide but\nsoon reappear -unchanged-. Except, that is, for one very curious fact:\nthe fast wave which has overtaken the slower one will have -jumped ahead-!\n\nQuite remarkably, one can obtain these "n-soliton" solutions -in closed\nform- by the same "inverse scattering method" first developed for studying\na nonlinear Schroedinger equation ("NLS" equation)! This discovery marked\nthe beginning of a beautiful subject, for the KdV equation is not the only\nnonlinear PDE exhibiting persistent solitary wave solutions, nor is it the\nonly one which is subject to solution by inverse scattering transformation\nmethods. Two more examples of such equations are the Liouville equation\n\n-u_(tt) + u_(xx) = exp(u)\n\nand the sine-Gordon equation\n\n-u_(tt) + u_(xx) = sin(u)\n\nBut there are many other examples, including certain "symmetry reductions"\nof the Einstein field equation (EFE); hence the term "gravitational\nsoliton", referring to certain exact solutions in gtr which model\n"colliding plane waves" and were obtained using methods from soliton\ntheory. (Even if you don\'t know what a gravitational wave is, or have any\nidea why their collisions are so interesting, you probably know what a\nblack hole is; it turns out that one way to attack the important problem\nof understanding the effect on the -interior- of a black hole of throwing\n"stuff" into the hole is by studying colliding plane waves. For example,\nthe interior of the famous Kerr vacuum is in fact "locally isometric" to a\ncertain colliding plane wave spacetime.)\n\nBefore I can sketch further reasons why the KdV equation and its relatives\nare so amazing, and so deserving of much closer study, I need to say\nsomething about a very general and useful tool for studying ordinary or\npartial differential equations (especially nonlinear ones), or systems of\nsame: "symmetry analysis". This vast program was first envisioned by\nSophus Lie, and extensively developed by him, his students, and by many\nmathematicians in our own time. It is of tremendous interest in its own\nright, both to theoretical and applied mathematicians and physicists, or\nindeed to anyone who has ever confronted the problem of solving a\ndifferential equation not discussed in elementary "cookbooks".\n\nTo understand the basic idea of symmetry analysis, suppose we have a PDE\nwritten in the form\n\nF(x,t,u,u_x,u_t,u_(xx)) = 0\n\ni.e. some function of the independent variables x,t, the dependent\nvariable u, and its derivatives u_x, u_t, u_(xx), etc., is required to\nvanish. In such a case, Lie defined the "point symmetry group" of the DE\nas the transformations (x,t,u) -> (X,T,U) which preserve the form of the\nequation. Thus, for the KdV, the symmetry group consists of\ntransformations (x,t,u) -> (X,T,U) such that transforming the original\nequation gives an equation having exactly the same form:\n\nU_T = U_(XXX) + U U_X\n\nLie gave an -algorithm- for determining the "point symmetries" of a DE (or\nsystem of DEs) by writing down a system of PDEs which are -linear- in\nhighest order derivatives involving the unknown coefficients xi,tau,phi.\nThis algorithm is -effective- in the sense that this system of\n"determining equations" can be solved by "triangularization". (At least,\nthis is true if-- as often happens--- Lie\'s system of "determining\nequations" for the given DE or system of DEs is "polynomially\ndifferential".) This process (relevant buzzwords include "Groebner basis",\n"differential algebra") is very similar to the Gaussian reduction of a\nsystem of linear equations.\n\n(The algorithm is effective, but carrying it out by hand--- just as for\nGaussian reduction--- can quickly become very tedious. Fortunately,\nsymbolic manipulation packages like Maple and Mathematica implement the\nnecessary computational techniques, making it possible to quickly carry\nout the most tedious parts of the computations. Again fortunately, once\none has a putative answer, one can easily check--- if necessary, "by\nhand", that the result is not incorrect.)\n\nIn the case of two independent variables x,t and one dependent\nvariable u, we seek a "flow"\n\nX = xi(x,t,u) @/@x + tau(x,t,u) @/@t + phi(x,t,u) @/@u\n\nwhose integral curves give the desired "symmetry transformations"\n\n(x,t,u) -> (X,T,U)\n\n(Similarly for PDEs or systems involving additional independent or\ndependent variables.) "Prolonging" the dependent variable to u_x, u_t,\nu_(xx),... we obtain the determining equations. Solving this system gives\nflows which generate, not the group itself, but its Lie algebra--- which\nis in many respects even more useful!\n\nFor example, the Lie algebra of the point symmetry group of the KdV\n\nu_t = u_(xxx) + u u_x\n\nturns out to have a basis, handed to us by our solution of the determining\nequations for the KdV, which consists of the following four vector fields\n\nX1 = @/@x (space translation)\n\nX2 = @/@t (time translation)\n\nX3 = t @/@x - @/@u\n\nX4 = x @/@x + 3 t @/@t - 2 u @/@u\n\nOne thing computing the point symmetry group does for us is that it\nenables us to immediately write down an Ansatz such as u(x,t) = f(kx-ct)\nwhich is -guaranteed- to reduce the number of variables; in this case,\nsuch "symmetry Ansatz" will reduce our nonlinear third order PDE to a\nnonlinear 3rd order ODE. (An arbitrary Ansatz is almost certain to turn\nout to be incompatible with a given PDE; the point is that guessing an\nAnsatz which will work is not always easy, although the traveling wave\nAnsatz will apply to any PDE which does not depend -explicitly- on\nor t.\n\nBut it turns out that (as Lie showed) ODEs also have symmetry groups, and\nwe can use a nontrivial symmetry group of any ODE (if one exists) to\nreduce the -order- of this ODE. For example, whenever one appeals (as I\ndid above) to an "integrating factor" to reduce the -order- of an ODE, one\nis secretly appealing Lie\'s methods!\n\nWhat all this means is that by quite elementary methods we can often find\n"particular" exact solutions to nonlinear PDEs, with properties "as\nordered". And with more work, we can sometimes obtain -general- solutions\nusing symmetry methods!\n\nHere are some easy examples of how the elementary "symmetry Ansatz" method\nworks, in the case of the KdV equation:\n\n1. the flow X = c X1 + k X2, where c, k are undetermined real constants,\nhas invariants\n\nc0 = kx-ct,\n\nc1 = u\n\nwhich leads to the symmetry Ansatz\n\nu(x,t) = f(kx-ct)\n\nPhysically speaking, writing down this Ansatz is of course the same thing\nas announcing our intention to look for -traveling wave solutions-, with\nwave number k ("crests per cm") and speed of propagation c ("crests per\nsec"). We already saw that in the case of the KdV, this does lead to\nperiodic solutions, the cnoidal waves, which obey a dispersion relation\nrelating k,c, and which in the limit of infinite period go over to the\n1-soliton solution modeling the quantitative observations of Scott\nRussell.\n\n2. the flow X3 = t @/@x - @/@u, where a is a real constant, has\ninvariants\n\nw = t\n\nz = u+x/t\n\nwhich leads to the symmetry Ansatz\n\nu(x,t) = f(t)-x/t\n\nPlugging this into the KdV and eliminating x gives the simple first order\nODE\n\nf(t)/t + f\'(t) = 0\n\nwhich has the solution\n\nf(t) = b/t\n\nor\n\nu(x,t) = (b-x)/t\n\nwhich is an example of a -rational- solution with a pole at t = 0. This\nis of course a very simple example, and because of the pole you might\nconsider this solution "unrealistic", but nonetheless, more complicated\nrational solutions are sometimes useful.\n\n3. the flow X4 = x @/@x + 3 t @/@t - 2 u @/@u has invariants\n\nc0 = x^3/t\n\nc1 = u x^2\n\nwhich we can rewrite\n\nw = x/t^(1/3) (new independent variable)\n\nz = u t^(2/3) (new dependent variable)\n\ngiving the symmetry Ansatz\n\nu(x,t) = t^(-2/3) f(x/t^(1/3)\n\nand the nonlinear non-autonomous third order ODE\n\n3 f\'\'\'(w) + (w + 3 f) f\' + 2 f = 0\n\nThis ODE admits the integrating factor w+f, yielding the nonlinear\nnon-autonomous second order ODE\n\n(3f + 3 w) f\'\' - 3 f\' - 3 (f\')^2/2 + f^3 + 2 2 f^2 + w^2 f\n\nwhich has no further obvious reduction (in fact, it should be the second\nPainleve transcendent), but which can be numerically integrated.\n\nBTW, if you have heard of the "Buckingham product theorem", the symmetry\nanalysis of Lie is a vast generalization of the idea of "similarity\nvariables" to include symmetries more complicated than simple scalings\nlike (x,t,u) -> (x^2,t^3,u^(-2)).\n\nLet me make some more observations concerning the KdV, whose origin will\nprobably initially appear very mysterious. If we rewrite the KdV in the\nform of a "divergence law"\n\n@/@t u + @/@x [-u_(xx) - u^2/2] = 0\n\nand form the integral\n\nM(t) = int_{-infty}^{infty} u(x,t) dx\n\nthen for -asymptotically vanishing solutions- of the KdV, i.e. those for\nwhich |u| -> 0 as |x| -> infty, such as the n-soliton solutions, we see\nthat M(t) is actually -constant- in time. Similarly, multiplying the KdV\nby u, we have\n\n@/@t [u^2/2] = u u_t\n\n= u u_(xxx) + u_x u_(xx) - u_x u_(xx) - u^2 u_x\n\n= @/@x [u u_(xx) - (u_x)^2 - u^3/3]\n\nwe find that the quantity\n\nP(t) = int_{-infty}^{infty} u(x,t)^/2 dx\n\nis constant for asymptotically vanishing solutions. With considerably\nmore ingenuity, we can find a third conserved quantity, the integral of\nu^3+u_x/2. These can be regarded, I claim, as respectively a "mass",\n"linear momentum", and "energy" characterizing a particular solution of\nthe KdV equation. In the case of a particular n-soliton solution of the\nKdV, we have a system of n solitonic entities, analogous to a Hamiltonian\nsystem (say) of n "particles", with the total mass M and so forth playing\nthe role of conserved quantities.\n\nThese claims raise several questions:\n\n1. Given a PDE, when do conserved density/flux pairs exist, and how can we\nfind them?\n\n2. When can we interpret them as things like energy and angular momentum?\n\n3. If we can interpret one density as "energy density" and another as\n"momentum density", when does our "energy flux" equal our "momentum\ndensity"?\n\nThese questions are at least partially answered in various books on\nsoliton theory (see citations below).\n\nNext, observe that from the mass M, energy E, and momentum P we obtain\nmore invariants like M^2 E -P, but we would not regard these trivial\nderived invariants as being in any sense "fundamental", or even very\ninteresting. The amazing thing is that the KdV (and many other equations\narising in soliton theory) in fact admits an -infinite hierarchy- of\n"fundamental" conservation laws! In other words, in addition to conserved\nquantities interpretable, at least by formal analogy, in terms of familiar\nphysical quantities like mass, linear momentum, angular momentum (in\nhigher dimensions), and energy, there is a whole infinite sequence of\nfurther conserved quantities which are not simply expressible in terms of\nearlier ones.\n\nLet me try to give some indication of how this works.\n\nAn elementary trick which can be employed in studying many "evolution\nPDEs" is to try to rewrite the original PDE in terms of a "potential".\nFor example, in the case of the KdV, we can write u = v_x where v(x,t)\nsatisfies the fourth-order PDE\n\nv_(xt) = v_x v_(xx) + v_(xxxx)\n\n(This is like rewriting the first order Maxwell equations in terms of\nsecond order potential equations.) We can of course now proceed to\ncompute the point symmetries of this equation using Lie\'s algorithm.\n\nBut to see why we might gain something by this procedure, observe that our\npotential equation arises as the Euler-Lagrange equation for a\n-Lagrangian-! To wit:\n\nL(x,t,v_x,v_t,v_(xx)) = [v_(xx)]^2/2 - [v_x]^3/6 + v_x v_t/2\n\nI stress that the ideas we are discussing work with Lagrangians of any\norder!\n\n(If have never seen the "Euler operator" for a Lagrangian as general as\nthis, it is:\n\nE[.] = @/@u - [ D_x @/@u_x + D_t @/@u_t ]\n\n+ [D_x^2 @/@u_(xx) + D_x D_y @/@u_(xy) + D_y^2 @/@u_(yy) ] - ...\n\nNotice the memorable alternating sum. Here, D_x and D_y are "total\nderivative operators". This means that when we take the partials @/@u_x\nand so forth, we treat u_x and the rest as variables, but when we take a\ntotal derivative of f(x,t,u), we must take account of the fact that u is\nactually a dependent variable depending on x,t, so D_x f = f_x + f_u u_x.)\n\nNext, following Hilbert, we can ask about the relationship between\ntransformations preserving the form of the Lagrangian, i.e.\ntransformations (x,t,v) -> (X,T,V) such that\n\nL -> [V_(XX)]^2/2 - [V_X]^3/6 + V_X V_T/2\n\nand the point symmetries of the Euler-Lagrange equation (the KdV potential\nequation), i.e. the transformations such that\n\nV_(XT) = V_X V_(XX) + V_(XXXX)\n\nNoether showed that this group of -variational symmetries- forms a\nsubgroup of the group of -point symmetries- (of the potential equation).\nEven better, the famous Noether Theorem tells us to construct a\n-conservation law- (for our potential v) from each variational symmetry!\n\nIn the course of her investigation into variational symmetries, Noether\nrealized that her Theorem actually applies to a much more general type of\ntransformation than Lie\'s point transformations. Namely, we can consider\nflows of form\n\nxi(x,t,v,v_x,..) @/@x + tau(x,t,v,v_x,..) @/@t + phi(x,t,v,v_x,..) @/@v\n\nUnlike Lie\'s point transformations, which act on a finite dimensional "jet\nspace" (in our example, the jet space comprises the variables\nx,t,v,v_x,v_(xt),v_(xx),v_(xxxx), but no derivatives of fifth or higher\norder), in general these "Lie/Baecklund transformations" (really due to\nNoether, but almost universally called after Lie and Baecklund, who worked\non related notions) act on the "infinite dimensional jet space" (x,t,u,\nand all its partial derivatives). As you might expect, it is harder to\ncompute the group of Lie/Baecklund symmetries than the group of point\nsymmetries, but if you can find some, they can often be used to find exact\nsolutions (much like point transformations). We can also use them to look\nfor "Baecklund morphisms", of which more in a moment. But the important\npoint here is that Noether\'s theorem gives a simple criterion for when a\nLie-Baecklund transformation is a generalized variational symmetry, and if\nso we obtain a conservation law!\n\nSo, one way of understanding what is going here is that the group of\nLie-Baecklund symmetries of the KdV potential equation is infinite\ndimensional, and most if not all of these give variational symmetries.\nEven better, it is possible to find "recursion operators" allowing us to\nrecursively compute, explicitly, the infinite list of generators of the\nLie algebra of this infinite dimensional group. This gives an infinite\nhierarchy of conservation laws for the potential v, and then using u =\n(D_x)^(-1) v, or written out more fully,\n\nu(x,t) = int_(-infty)^x v(w,t) dw\n\nwe can write down a corresponding infinite hierarchy of conservation laws\nfor the KdV itself!\n\n(The historical genesis of Noether\'s theorem is quite interesting.\nDuring the early development of gtr, Hilbert needed to better understand\nthe divergence identity satisfied by the Einstein tensor and how it\nrelated to Hilbert\'s "variational reformulation" of the EFE, and he\ndiscussed his confusion with Noether. Her resolution turned out to be far\nmore important than the original question! The relationship between\nsymmetry and conservation laws which she discovered is of course\nfundamental to all of modern physics, and is now regarded as one of her\ngreatest discoveries. Unfortunately, the general version--- due to\nNoether herself--- which is valid for nonquadratic Lagrangians and even\nfor Lie-Baecklund symmetries, is apparently not known to many physicists.)\n\nI just said that Lie-Baecklund symmetries can be used to hunt for\nBaecklund morphisms. These are mappings from the solution space of one\npde F(x,t,u,u_x..) to that of another pde G(x,t,v,v_x,...), which is\nusually of the same order. They are interesting and useful for many\nreasons, among them: sometimes we can map solutions of a -linear- pde to\nsolutions of a -nonlinear- one! (This happens, for example, with the\nsine-Gordon equation mentioned above.)\n\nIt turns out that there is a "Baecklund homomorphism" mapping the solution\nspace of the KdV into the solution space of another PDE, sometimes called\nthe Miura equation\n\nu_t = u_(xxx) + u^2 u_x\n\nbut I won\'t try to explain here why this is interesting; rather, let me\njust mention one other random fact about the KdV:\n\nWe saw that the KdV potential arises from a Lagrangian. It turns out that\nthe KdV equation can also be given -Hamiltonian- form! Indeed, this can be\ndone in -two- distinct ways!\n\nThis turns out to be closely related to the existence of an infinite\nhierarchy of conservation laws. As you probably know, one of Poincare\'s\ngreatest discoveries was the fact that, in some sense, "most" Hamiltonian\nsystems are not solvable in closed form (e.g. the -general- three body\nproblem of Newtonian gravitation). The KdV, OTH, admits infinitely many\n"integrals of motion" (corresponding to the conserved quantities mentioned\nabove). This is characteristic of a "completely integrable Hamiltonian\nsystem". (See again my remarks above concerning n-soliton solutions of\nthe KdV and conserved quantities.)\n\n> and that Russell discovered the relation between speed v, amplitude A,\n> and width lambda of a solitonic bump.\n\nRussell was studying surface waves in water contained in a straight sided\nshallow trough with a flat bottom; his procedure was to drop a weight at\none end of the trough. (He was trying to reproduce the "persistent\ndisturbance" he had observed "in Nature" during a famous horseback chase\nalong an English canal.)\n\nLater, Korteweg and de Vries were able to find a theoretical model of\nshallow water waves, namely their now famous equation, which explained\nRussell\'s observations. The sech^2 solution described above corresponds\nwell to the shape observed by Russell.\n\nBTW, you might have noticed that the KdV equation only admits\nunidirectional propagation! This is of course an artifact of our choice\nof governing equation; an appropriate fourth order PDE, the "Boussinesq\nequation", admits periodic waves (and persistent solitary waves) traveling\nin either direction. There are also higher dimensional generalizations of\nthe KdV equation (the best known is called the "Kadomtsev/Petviashvili\nequation").\n\n> He pretended (wrongly, I think) that this relation is given by\n>\n> v = 2A - 4/lambda^2\n>\n> (there is a dimensional problem)\n> when the soliton local amplitude w is given as\n>\n> w(x,t)= A / cosh^2[(x-vt)/lambda]\n\nThis should fall out from a careful analysis of the above mentioned\ndispersion relation for the cnoidal waves when you let the period tend to\ninfinity. You\'ll need to define "lambda" as a combination of other\nvariables (see the expression for the 1-soliton which I gave above).\nNormally one "nondimensionalizes" PDEs for analytical convenience; the\nappropriate numerical factors can be reinserted later. The KdV as I wrote\nit has been nondimensionalized.\n\nRegarding Russell\'s formula, your friend was probably reading the section\non the KdV equation in the classic textbook\n\nauthor = {Horace Lamb},\ntitle = {Hydrodynamics},\npublisher = {Dover},\nyear = 1945}\n\nwhich has a clear account of the simplifications involved in deriving the\nKdV equation. This is an inexpensive reprint of the sixth edition (1932)\nof a classic first published in 1879; since soliton theory only began to\nemerge in the 1960s, Lamb had a good excuse for what would otherwise be an\nintolerable sin of omission :-/ Nonetheless this book remains a gold mine\nof useful information about nineteenth century work on hydrodynamics,\nwhich remains fundamental.\n\nFor more about solitons, try\n\nauthor = {P.G. Drazin and R.S. Johnson},\ntitle = {Solitons : an introduction},\npublisher = {Cambridge University Press},\nyear = 1989}\n\neditor = {Gu Chaohao},\ntitle = {Soliton Theory and Its Applications},\npublisher = {Springer-Verlag},\nyear = 1990}\n\nauthor = {G. L. Lamb},\ntitle = {Elements of soliton theory},\npublisher = {Wiley},\nyear = 1980}\n\nThe book by Drazin and Johnson has an excellent discussion of the above\nnoted opposing tendencies of nonlinear steepening and dispersive\nflattening. The first article in the book edited by Chaohao has some\npictures of phase plane analysis for some interesting variants of the KdV\nand a good discussion of some important areas of mathematical physics and\nbiology where solitons arise. (A few random buzzwords: Toda lattice,\ninstantons, "wetware memory".)\n\n"Dynamical system techniques", such as phase portraits for autonomous\nsystems of ODEs, are discussed in many, many textbooks. One I\nparticularly like is\n\nauthor = {Ferdinand Verhulst},\ntitle = {Nonlinear differential equations and dynamical systems},\npublisher = {Springer-Verlag},\nedition = {second},\nyear = 1996}\n\nFor the properties of the Jacobi elliptic functions (including their\ndouble periodicity) and their use in solving nonlinear ODEs, see\n\nauthor = {Harold T. Davis},\ntitle = {Introduction to nonlinear differential and integral equations},\npublisher = {Dover},\nyear = 1962}\n\nFor Lie analysis of PDEs (including the KdV equation, sine-Gordon\nequation, and Liouville equation), see (in approximate order of\ndifficulty)\n\nauthor = {Lawrence Dresner},\ntitle = {Applications of {L}ie\'s theory of ordinary and partial\ndifferential\nequations},\npublisher = {IOP Publishing},\nyear = 1999}\n\nauthor = {Brian J. Cantwell},\ntitle = {Introduction to symmetry analysis},\npublisher = {Cambridge University Press},\nyear = 2002}\n\nauthor = {Bluman, George W., and Kumei, Sukeyuki},\ntitle = {Symmetries and Differential Equations},\nseries = {Applied mathematical sciences},\nvolume =81,\npublisher = {Springer-Verlag},\nyear = {1989}}\n\nauthor = {Peter J. Olver},\ntitle = {Applications of {L}ie Groups to Differential Equations},\nseries = {Graduate Texts in Mathematics},\nvolume = 107,\npublisher = {Springer-Verlag},\nyear = 1993}\n\nThe first of these is a sketchy, but gives a very nice introduction to\nsome of the basic ideas. The last three books all offer a fairly\ncomprehensive introduction to the symmetry analysis of PDEs, including\ncomputing the Lie groups of point symmetries, Lie-Baecklund symmetries,\nvariational symmetries, Noether\'s Theorem, "recursion operators", and the\ninfinite hierarchy of Lie-Baeklund symmetries and conservation laws for\nthe KdV. Olver also discusses completely integrable Hamiltonian systems.\n\nFor Baecklund transformations, try the article in Chaohao (the book cited\nabove) or\n\nauthor = {C. Rogers and W.K. Schief},\ntitle = {Baecklund and {D}arboux transformations:\ngeometry and modern applications in soliton theory},\npublisher = {Cambridge University Press},\nseries = {Cambridge texts in applied mathematics},\nyear = 2002}\n\nFor nonlinear dispersion relations, see the article by Whitham in\n\neditor = {A. H. Taub},\ntitle = {Studies in Applied Mathematics},\npublisher = {Mathematical Association of America},\nseries = {Studies in Mathematics},\nvolume = 7,\nyear = 1971}\n\nFor scattering theory, see the article by Kata in that book, or the\nrelevant article in Chaohao.\n\nSpeaking of Taub and Stephani, who are well known for their work in gtr:\nas I mentioned above, various methods of soliton theory have been applied\nto find new solutions of the EFE, and gtr enthusiasts will want to see the\nrelevant article in Chaohao or the book\n\nauthor = {V. Belinski and E. Verdaguer},\ntitle = {Gravitational solitons},\npublisher = {Cambridge University Press},\nseries = {Cambridge monographs on mathematical physics},\nyear = 2001}\n\n"T. Essel" (hiding somewhere in cyberspace)\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Fri, 28 May 2004, Francois Belfort wrote:
> An acquaintance told me that all one needs to know about solitons is
> that they are a single travelling bumps,
The UW library catalog lists 72 books on solitons; in fact, there are
-many- things concerning solitons which are well worth learning! I'll try
to sketch a few of them in this post.
The particular equation you asked about is probably the most famous
example of a PDE exhibiting soliton solutions, the "Korteweg/de Vries
equation", or "KdV equation" for short. This can be written
u_t = u u_x + u_(xxx)
(The form you gave is just an alternative "normalization"; you can easily
find a coordinate transformation taking that form into this one.)
A fundamental point concerning the KdV equation is that it exhibits two
opposing tendencies:
1. "nonlinear convection", which tends to -steepen- wavecrests,
2. "dispersion", which tends to -flatten- wave crests.
The idea is that by carefully balancing these two effects, it is possible
to have a traveling wave, with a single wavecrest, which propagates
without changing shape or losing energy. This is called a "solitary
wave".
To understand this better, we should attempt to find the general traveling
wave solution of the KdV equation. To do this, we plug the traveling wave
Ansatz
u(x,t) = f(x-ct)
into the KdV equation (more about this later) , obtaining the ODE
f'''(w) = (c - f(w)) f'(w)
This admits the integrating factor f(w), so we can reduce it to a second
order autonomous (nonlinear) ODE:
f''(w) + f(w)^/2 - c f(w) + b =
Let us study the phase portrait of this "once-reduced" ODE. We find that
there are two critical points, at
{ (0,0) =[/itex] saddle
(f,f') = {
{ (2c,0) = center
Note that in this phase portrait, we have a -sepatrix- which starts at the
origin, winds once around the other critical point, and returns to the
origin. Inside this sepatrix we have trajectories which are diffeomorphic
to ellipses but "sharper" at their left hand ends. These correspond to
-periodic solutions- of our ODE--- i.e. to periodic traveling wavetrains.
These periodic solutions may be written explicitly in terms of the Jacobi
elliptic function cn(z; k) in the form
u(x,t) = A cn(B (x-ct) + C, k)^2
For this reason, they were called "cnoidal waves" by Korteweg and de
Vries, who discovered them. As they noted, the period may expressed as an
elliptic integral. Graphically, cnoidal waves look much like cosine
waves, but close examination shows they have sharper crests and flatter
troughs. Indeed, the function cn(.,.) also arises in solving the pendulum
equation, whose linearization is the familiar harmonic oscillator
equation, so their appearance here (and in many other places in the study
of nonlinear DEs) is not too surprising. Using standard facts about
Jacobi elliptic functions, one can find an explicit "dispersion relation"
between the wavenumber and speed of the periodic traveling waves---
working out the details is a good exercise in -nonlinear dispersion-.
As the period goes to infinity, the corresponding trajectories approach
the sepatrix from inside, which of course is a -non-periodic- solution.
It corresponds to the famous "one crested soliton" or "1-soliton" solution
of the KdV equation:
u(x,t) = 3 c sech^2 (\sqrt(c)/2 (x-ct) + e)
This solution represents a disturbance with a single crest which travels
indefinitely without changing shape, so it is a "solitary wave". But
saying it is a "soliton" implies much more! To see why, observe that from
what we've said so far, we might well expect that our solitary wave is
-unstable-, i.e. that small perturbations should either change it into a
long period cnoidal wave, or into a disturbance which blows up.
Alternatively, because this solitary wave arises from a governing equation
which balances the tendency of nonlinear convection to steepen wave crests
against the tendency of dispersion to flatten them (because any frequency
components will travel at different speeds, thus dispersing an initial
wave profile), we might well expect solitary waves to be unstable---
surely this "balancing" is a delicate affair!
But in fact, decades of work (both experimental and theoretical, beginning
with Russell's work) show that solitons are stable against small
-nonlinear- perturbations. This is, or should be, quite surprising!
The remarkable stability of solitons is well illustrated by their very
surprising nonlinear interactions. It turns out that we can find explicit
"n-soliton" solutions to the KdV, with n crests of differing heights and
speeds; these solitary waves interact nonlinearly when they collide but
soon reappear -unchanged-. Except, that is, for one very curious fact:
the fast wave which has overtaken the slower one will have -jumped ahead-!
Quite remarkably, one can obtain these "n-soliton" solutions -in closed
form- by the same "inverse scattering method" first developed for studying
a nonlinear Schroedinger equation ("NLS" equation)! This discovery marked
the beginning of a beautiful subject, for the KdV equation is not the only
nonlinear PDE exhibiting persistent solitary wave solutions, nor is it the
only one which is subject to solution by inverse scattering transformation
methods. Two more examples of such equations are the Liouville equation
-u_(tt) + u_(xx) = \exp(u)
and the sine-Gordon equation
-u_(tt) + u_(xx) = sin(u)
But there are many other examples, including certain "symmetry reductions"
of the Einstein field equation (EFE); hence the term "gravitational
soliton", referring to certain exact solutions in gtr which model
"colliding plane waves" and were obtained using methods from soliton
theory. (Even if you don't know what a gravitational wave is, or have any
idea why their collisions are so interesting, you probably know what a
black hole is; it turns out that one way to attack the important problem
of understanding the effect on the -interior- of a black hole of throwing
"stuff" into the hole is by studying colliding plane waves. For example,
the interior of the famous Kerr vacuum is in fact "locally isometric" to a
certain colliding plane wave spacetime.)
Before I can sketch further reasons why the KdV equation and its relatives
are so amazing, and so deserving of much closer study, I need to say
something about a very general and useful tool for studying ordinary or
partial differential equations (especially nonlinear ones), or systems of
same: "symmetry analysis". This vast program was first envisioned by
Sophus Lie, and extensively developed by him, his students, and by many
mathematicians in our own time. It is of tremendous interest in its own
right, both to theoretical and applied mathematicians and physicists, or
indeed to anyone who has ever confronted the problem of solving a
differential equation not discussed in elementary "cookbooks".
To understand the basic idea of symmetry analysis, suppose we have a PDE
written in the form
F(x,t,u,u_x,u_t,u_(xx)) =
i.e. some function of the independent variables x,t, the dependent
variable u, and its derivatives u_x, u_t, u_(xx), etc., is required to
vanish. In such a case, Lie defined the "point symmetry group" of the DE
as the transformations (x,t,u) -> (X,T,U) which preserve the form of the
equation. Thus, for the KdV, the symmetry group consists of
transformations (x,t,u) -> (X,T,U) such that transforming the original
equation gives an equation having exactly the same form:
U_T = U_(XXX) + U U_X
Lie gave an -algorithm- for determining the "point symmetries" of a DE (or
system of DEs) by writing down a system of PDEs which are -linear- in
highest order derivatives involving the unknown coefficients \xi,\tau,\phi.
This algorithm is -effective- in the sense that this system of
"determining equations" can be solved by "triangularization". (At least,
this is true if-- as often happens--- Lie's system of "determining
equations" for the given DE or system of DEs is "polynomially
differential".) This process (relevant buzzwords include "Groebner basis",
"differential algebra") is very similar to the Gaussian reduction of a
system of linear equations.
(The algorithm is effective, but carrying it out by hand--- just as for
Gaussian reduction--- can quickly become very tedious. Fortunately,
symbolic manipulation packages like Maple and Mathematica implement the
necessary computational techniques, making it possible to quickly carry
out the most tedious parts of the computations. Again fortunately, once
one has a putative answer, one can easily check--- if necessary, "by
hand", that the result is not incorrect.)
In the case of two independent variables x,t and one dependent
variable u, we seek a "flow"
X = \xi(x,t,u) @/@x + \tau(x,t,u) @/@t + \phi(x,t,u) @/@u
whose integral curves give the desired "symmetry transformations"
(x,t,u) -> (X,T,U)
(Similarly for PDEs or systems involving additional independent or
dependent variables.) "Prolonging" the dependent variable to u_x, u_t,u_(xx),... we obtain the determining equations. Solving this system gives
flows which generate, not the group itself, but its Lie algebra--- which
is in many respects even more useful!
For example, the Lie algebra of the point symmetry group of the KdV
u_t = u_(xxx) + u u_x
turns out to have a basis, handed to us by our solution of the determining
equations for the KdV, which consists of the following four vector fields
X1 = @/@x (space translation)
X2 = @/@t (time translation)
X3 = t @/@x - @/@uX4 = x @/@x + 3 t @/@t - 2 u @/@u
One thing computing the point symmetry group does for us is that it
enables us to immediately write down an Ansatz such as u(x,t) = f(kx-ct)
which is -guaranteed- to reduce the number of variables; in this case,
such "symmetry Ansatz" will reduce our nonlinear third order PDE to a
nonlinear 3rd order ODE. (An arbitrary Ansatz is almost certain to turn
out to be incompatible with a given PDE; the point is that guessing an
Ansatz which will work is not always easy, although the traveling wave
Ansatz will apply to any PDE which does not depend -explicitly- on
or t.
But it turns out that (as Lie showed) ODEs also have symmetry groups, and
we can use a nontrivial symmetry group of any ODE (if one exists) to
reduce the -order- of this ODE. For example, whenever one appeals (as I
did above) to an "integrating factor" to reduce the -order- of an ODE, one
is secretly appealing Lie's methods!
What all this means is that by quite elementary methods we can often find
"particular" exact solutions to nonlinear PDEs, with properties "as
ordered". And with more work, we can sometimes obtain -general- solutions
using symmetry methods!
Here are some easy examples of how the elementary "symmetry Ansatz" method
works, in the case of the KdV equation:
1. the flow X = c X1 + k X2, where c, k are undetermined real constants,
has invariants
c0 = kx-ct,c1 = u
which leads to the symmetry Ansatz
u(x,t) = f(kx-ct)
Physically speaking, writing down this Ansatz is of course the same thing
as announcing our intention to look for -traveling wave solutions-, with
wave number k ("crests per cm") and speed of propagation c ("crests per
sec"). We already saw that in the case of the KdV, this does lead to
periodic solutions, the cnoidal waves, which obey a dispersion relation
relating k,c, and which in the limit of infinite period go over to the
1-soliton solution modeling the quantitative observations of Scott
Russell.
2. the flow X3 = t @/@x - @/@u, where a is a real constant, has
invariants
w = tz = u+x/t
which leads to the symmetry Ansatz
u(x,t) = f(t)-x/t
Plugging this into the KdV and eliminating x gives the simple first order
ODE
f(t)/t + f'(t) =
which has the solution
f(t) = b/t
or
u(x,t) = (b-x)/t
which is an example of a -rational- solution with a pole at t = . This
is of course a very simple example, and because of the pole you might
consider this solution "unrealistic", but nonetheless, more complicated
rational solutions are sometimes useful.
3. the flow X4 = x @/@x + 3 t @/@t - 2 u @/@u has invariants
c0 = x^3/tc1 = u x^2
which we can rewrite
w = x/t^(1/3) (new independent variable)
z = u t^(2/3) (new dependent variable)
giving the symmetry Ansatz
u(x,t) = t^(-2/3) f(x/t^(1/3)
and the nonlinear non-autonomous third order ODE
3 f'''(w) + (w + 3 f) f' + 2 f =
This ODE admits the integrating factor w+f, yielding the nonlinear
non-autonomous second order ODE
(3f + 3 w) f'' - 3 f' - 3 (f')^2/2 + f^3 + 2 2 f^2 + w^2 f
which has no further obvious reduction (in fact, it should be the second
Painleve transcendent), but which can be numerically integrated.
BTW, if you have heard of the "Buckingham product theorem", the symmetry
analysis of Lie is a vast generalization of the idea of "similarity
variables" to include symmetries more complicated than simple scalings
like (x,t,u) -> (x^2,t^3,u^(-2)).
Let me make some more observations concerning the KdV, whose origin will
probably initially appear very mysterious. If we rewrite the KdV in the
form of a "divergence law"
@/@t u + @/@x [-u_(xx) - u^2/2] =
and form the integral
M(t) = \int_{-\infty}^{\infty} u(x,t) dx
then for -asymptotically vanishing solutions- of the KdV, i.e. those for
which |u| -> as |x| -> \infty, such as the n-soliton solutions, we see
that M(t) is actually -constant- in time. Similarly, multiplying the KdV
by u, we have
@/@t [u^2/2] = u u_t= u u_(xxx) + u_x u_(xx) - u_x u_(xx) - u^2 u_x= @/@x [u u_(xx) - (u_x)^2 - u^3/3]
we find that the quantity
P(t) = \int_{-\infty}^{\infty} u(x,t)^/2 dx
is constant for asymptotically vanishing solutions. With considerably
more ingenuity, we can find a third conserved quantity, the integral of
u^3+u_x/2. These can be regarded, I claim, as respectively a "mass",
"linear momentum", and "energy" characterizing a particular solution of
the KdV equation. In the case of a particular n-soliton solution of the
KdV, we have a system of n solitonic entities, analogous to a Hamiltonian
system (say) of n "particles", with the total mass M and so forth playing
the role of conserved quantities.
These claims raise several questions:
1. Given a PDE, when do conserved density/flux pairs exist, and how can we
find them?
2. When can we interpret them as things like energy and angular momentum?
3. If we can interpret one density as "energy density" and another as
"momentum density", when does our "energy flux" equal our "momentum
density"?
These questions are at least partially answered in various books on
soliton theory (see citations below).
Next, observe that from the mass M, energy E, and momentum P we obtain
more invariants like M^2 E -P, but we would not regard these trivial
derived invariants as being in any sense "fundamental", or even very
interesting. The amazing thing is that the KdV (and many other equations
arising in soliton theory) in fact admits an -infinite hierarchy- of
"fundamental" conservation laws! In other words, in addition to conserved
quantities interpretable, at least by formal analogy, in terms of familiar
physical quantities like mass, linear momentum, angular momentum (in
higher dimensions), and energy, there is a whole infinite sequence of
further conserved quantities which are not simply expressible in terms of
earlier ones.
Let me try to give some indication of how this works.
An elementary trick which can be employed in studying many "evolution
PDEs" is to try to rewrite the original PDE in terms of a "potential".
For example, in the case of the KdV, we can write u = v_x where v(x,t)
satisfies the fourth-order PDE
v_(xt) = v_x v_(xx) + v_(xxxx)
(This is like rewriting the first order Maxwell equations in terms of
second order potential equations.) We can of course now proceed to
compute the point symmetries of this equation using Lie's algorithm.
But to see why we might gain something by this procedure, observe that our
potential equation arises as the Euler-Lagrange equation for a
-Lagrangian-! To wit:
L(x,t,v_x,v_t,v_(xx)) = [v_(xx)]^2/2 - [v_x]^3/6 + v_x v_t/2
I stress that the ideas we are discussing work with Lagrangians of any
order!
(If have never seen the "Euler operator" for a Lagrangian as general as
this, it is:
E[.] = @/@u - [ D_x @/@u_x + D_t @/@u_t ]+ [D_x^2 @/@u_(xx) + D_x D_y @/@u_(xy) + D_y^2 @/@u_(yy) ] - ...
Notice the memorable alternating sum. Here, D_x and D_y are "total
derivative operators". This means that when we take the partials @/@u_x
and so forth, we treat u_x and the rest as variables, but when we take a
total derivative of f(x,t,u), we must take account of the fact that u is
actually a dependent variable depending on x,t, so D_x f = f_x + f_u u_x.)
Next, following Hilbert, we can ask about the relationship between
transformations preserving the form of the Lagrangian, i.e.
transformations (x,t,v) -> (X,T,V) such that
L -> [V_(XX)]^2/2 - [V_X]^3/6 + V_X V_T/2
and the point symmetries of the Euler-Lagrange equation (the KdV potential
equation), i.e. the transformations such that
V_(XT) = V_X V_(XX) + V_(XXXX)
Noether showed that this group of -variational symmetries- forms a
subgroup of the group of -point symmetries- (of the potential equation).
Even better, the famous Noether Theorem tells us to construct a
-conservation law- (for our potential v) from each variational symmetry!
In the course of her investigation into variational symmetries, Noether
realized that her Theorem actually applies to a much more general type of
transformation than Lie's point transformations. Namely, we can consider
flows of form
\xi(x,t,v,v_x,..) @/@x + \tau(x,t,v,v_x,..) @/@t + \phi(x,t,v,v_x,..) @/@v
Unlike Lie's point transformations, which act on a finite dimensional "jet
space" (in our example, the jet space comprises the variables
x,t,v,v_x,v_(xt),v_(xx),v_(xxxx), but no derivatives of fifth or higher
order), in general these "Lie/Baecklund transformations" (really due to
Noether, but almost universally called after Lie and Baecklund, who worked
on related notions) act on the "infinite dimensional jet space" (x,t,u,
and all its partial derivatives). As you might expect, it is harder to
compute the group of Lie/Baecklund symmetries than the group of point
symmetries, but if you can find some, they can often be used to find exact
solutions (much like point transformations). We can also use them to look
for "Baecklund morphisms", of which more in a moment. But the important
point here is that Noether's theorem gives a simple criterion for when a
Lie-Baecklund transformation is a generalized variational symmetry, and if
so we obtain a conservation law!
So, one way of understanding what is going here is that the group of
Lie-Baecklund symmetries of the KdV potential equation is infinite
dimensional, and most if not all of these give variational symmetries.
Even better, it is possible to find "recursion operators" allowing us to
recursively compute, explicitly, the infinite list of generators of the
Lie algebra of this infinite dimensional group. This gives an infinite
hierarchy of conservation laws for the potential v, and then using u =
(D_x)^(-1) v, or written out more fully,
u(x,t) = \int_(-\infty)^x v(w,t) dw
we can write down a corresponding infinite hierarchy of conservation laws
for the KdV itself!
(The historical genesis of Noether's theorem is quite interesting.
During the early development of gtr, Hilbert needed to better understand
the divergence identity satisfied by the Einstein tensor and how it
related to Hilbert's "variational reformulation" of the EFE, and he
discussed his confusion with Noether. Her resolution turned out to be far
more important than the original question! The relationship between
symmetry and conservation laws which she discovered is of course
fundamental to all of modern physics, and is now regarded as one of her
greatest discoveries. Unfortunately, the general version--- due to
Noether herself--- which is valid for nonquadratic Lagrangians and even
for Lie-Baecklund symmetries, is apparently not known to many physicists.)
I just said that Lie-Baecklund symmetries can be used to hunt for
Baecklund morphisms. These are mappings from the solution space of one
pde F(x,t,u,u_x..) to that of another pde G(x,t,v,v_x,...), which is
usually of the same order. They are interesting and useful for many
reasons, among them: sometimes we can map solutions of a -linear- pde to
solutions of a -nonlinear- one! (This happens, for example, with the
sine-Gordon equation mentioned above.)
It turns out that there is a "Baecklund homomorphism" mapping the solution
space of the KdV into the solution space of another PDE, sometimes called
the Miura equation
u_t = u_(xxx) + u^2 u_x
but I won't try to explain here why this is interesting; rather, let me
just mention one other random fact about the KdV:
We saw that the KdV potential arises from a Lagrangian. It turns out that
the KdV equation can also be given -Hamiltonian- form! Indeed, this can be
done in -two- distinct ways!
This turns out to be closely related to the existence of an infinite
hierarchy of conservation laws. As you probably know, one of Poincare's
greatest discoveries was the fact that, in some sense, "most" Hamiltonian
systems are not solvable in closed form (e.g. the -general- three body
problem of Newtonian gravitation). The KdV, OTH, admits infinitely many
"integrals of motion" (corresponding to the conserved quantities mentioned
above). This is characteristic of a "completely integrable Hamiltonian
system". (See again my remarks above concerning n-soliton solutions of
the KdV and conserved quantities.)
> and that Russell discovered the relation between speed v, amplitude A,
> and width \lambda of a solitonic bump.
Russell was studying surface waves in water contained in a straight sided
shallow trough with a flat bottom; his procedure was to drop a weight at
one end of the trough. (He was trying to reproduce the "persistent
disturbance" he had observed "in Nature" during a famous horseback chase
along an English canal.)
Later, Korteweg and de Vries were able to find a theoretical model of
shallow water waves, namely their now famous equation, which explained
Russell's observations. The sech^2 solution described above corresponds
well to the shape observed by Russell.
BTW, you might have noticed that the KdV equation only admits
unidirectional propagation! This is of course an artifact of our choice
of governing equation; an appropriate fourth order PDE, the "Boussinesq
equation", admits periodic waves (and persistent solitary waves) traveling
in either direction. There are also higher dimensional generalizations of
the KdV equation (the best known is called the "Kadomtsev/Petviashvili
equation").
> He pretended (wrongly, I think) that this relation is given by
>
> [itex]v = 2A - 4/\lambda^2
>
> (there is a dimensional problem)
> when the soliton local amplitude w is given as
>
> w(x,t)= A / cosh^2[(x-vt)/\lambda]
This should fall out from a careful analysis of the above mentioned
dispersion relation for the cnoidal waves when you let the period tend to
infinity. You'll need to define "\lambda" as a combination of other
variables (see the expression for the 1-soliton which I gave above).
Normally one "nondimensionalizes" PDEs for analytical convenience; the
appropriate numerical factors can be reinserted later. The KdV as I wrote
it has been nondimensionalized.
Regarding Russell's formula, your friend was probably reading the section
on the KdV equation in the classic textbook
author = {Horace Lamb},
title = {Hydrodynamics},
publisher = {Dover},
year = 1945}
which has a clear account of the simplifications involved in deriving the
KdV equation. This is an inexpensive reprint of the sixth edition (1932)
of a classic first published in 1879; since soliton theory only began to
emerge in the 1960s, Lamb had a good excuse for what would otherwise be an
intolerable sin of omission :-/ Nonetheless this book remains a gold mine
of useful information about nineteenth century work on hydrodynamics,
which remains fundamental.
For more about solitons, try
author = {P.G. Drazin and R.S. Johnson},
title = {Solitons : an introduction},
publisher = {Cambridge University Press},
year = 1989}
editor = {Gu Chaohao},
title = {Soliton Theory and Its Applications},
publisher = {Springer-Verlag},
year = 1990}
author = {G. L. Lamb},
title = {Elements of soliton theory},
publisher = {Wiley},
year = 1980}
The book by Drazin and Johnson has an excellent discussion of the above
noted opposing tendencies of nonlinear steepening and dispersive
flattening. The first article in the book edited by Chaohao has some
pictures of phase plane analysis for some interesting variants of the KdV
and a good discussion of some important areas of mathematical physics and
biology where solitons arise. (A few random buzzwords: Toda lattice,
instantons, "wetware memory".)
"Dynamical system techniques", such as phase portraits for autonomous
systems of ODEs, are discussed in many, many textbooks. One I
particularly like is
author = {Ferdinand Verhulst},
title = {Nonlinear differential equations and dynamical systems},
publisher = {Springer-Verlag},
edition = {second},
year = 1996}
For the properties of the Jacobi elliptic functions (including their
double periodicity) and their use in solving nonlinear ODEs, see
author = {Harold T. Davis},
title = {Introduction to nonlinear differential and integral equations},
publisher = {Dover},
year = 1962}
For Lie analysis of PDEs (including the KdV equation, sine-Gordon
equation, and Liouville equation), see (in approximate order of
difficulty)
author = {Lawrence Dresner},
title = {Applications of {L}ie's theory of ordinary and partial
differential
equations},
publisher = {IOP Publishing},
year = 1999}
author = {Brian J. Cantwell},
title = {Introduction to symmetry analysis},
publisher = {Cambridge University Press},
year = 2002}
author = {Bluman, George W., and Kumei, Sukeyuki},
title = {Symmetries and Differential Equations},
series = {Applied mathematical sciences},
volume =81,
publisher = {Springer-Verlag},
year = {1989}}
author = {Peter J. Olver},
title = {Applications of {L}ie Groups to Differential Equations},
series = {Graduate Texts in Mathematics},
volume = 107,
publisher = {Springer-Verlag},
year = 1993}
The first of these is a sketchy, but gives a very nice introduction to
some of the basic ideas. The last three books all offer a fairly
comprehensive introduction to the symmetry analysis of PDEs, including
computing the Lie groups of point symmetries, Lie-Baecklund symmetries,
variational symmetries, Noether's Theorem, "recursion operators", and the
infinite hierarchy of Lie-Baeklund symmetries and conservation laws for
the KdV. Olver also discusses completely integrable Hamiltonian systems.
For Baecklund transformations, try the article in Chaohao (the book cited
above) or
author = {C. Rogers and W.K. Schief},
title = {Baecklund and {D}arboux transformations:
geometry and modern applications in soliton theory},
publisher = {Cambridge University Press},
series = {Cambridge texts in applied mathematics},
year = 2002}
For nonlinear dispersion relations, see the article by Whitham in
editor = {A. H. Taub},
title = {Studies in Applied Mathematics},
publisher = {Mathematical Association of America},
series = {Studies in Mathematics},
volume = 7,
year = 1971}
For scattering theory, see the article by Kata in that book, or the
relevant article in Chaohao.
Speaking of Taub and Stephani, who are well known for their work in gtr:
as I mentioned above, various methods of soliton theory have been applied
to find new solutions of the EFE, and gtr enthusiasts will want to see the
relevant article in Chaohao or the book
author = {V. Belinski and E. Verdaguer},
title = {Gravitational solitons},
publisher = {Cambridge University Press},
series = {Cambridge monographs on mathematical physics},
year = 2001}
"T. Essel" (hiding somewhere in cyberspace)
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\ntessel@tum.bot writes\n\n<random quote from a mindblowing post>\n\n>I just said that Lie-Baecklund symmetries can be used to hunt for\n>Baecklund morphisms. These are mappings from the solution space of one\n>pde F(x,t,u,u_x..) to that of another pde G(x,t,v,v_x,...), which is\n>usually of the same order. They are interesting and useful for many\n>reasons, among them: sometimes we can map solutions of a -linear- pde to\n>solutions of a -nonlinear- one! (This happens, for example, with the\n>sine-Gordon equation mentioned above.)\n\nAm I to infer that work on solitons as a potential explanation for\nelementary particles has not ceased, but continues and progresses to\nthis day?\n\nUnfortunately I only really understood about 0.1% of the post (hey, I\'m\nan optimist) but I got the impression that solitons could be generated\nin mathematical structures similar to those in common use in theoretical\nphysics.\n\nI am also intrigued by several statements, one of which I quoted above.\nIf one can map solutions of a linear pde to a non-linear one then in a\nworld where you only ever see solutions it would be easy to assume they\nwere modelled by a linear one and not (which could be the case) the non-\nlinear one. Given the simplifications allowed by linear anything\ncompared to non-linear (which tend towards being intractable) one could\nbe rather badly sidetracked at quite an early point.\n\n--\nOz\nThis post is worth absolutely nothing and is probably fallacious.\n\nBTOPENWORLD address about to cease. DEMON address no longer in use.\n>>Use oz@farmeroz.port995.com (whitelist check on first posting)<<\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>tessel@tum.bot writes
<random quote from a mindblowing post>
>I just said that Lie-Baecklund symmetries can be used to hunt for
>Baecklund morphisms. These are mappings from the solution space of one
>pde F(x,t,u,u_x..) to that of another pde G(x,t,v,v_x,...), which is
>usually of the same order. They are interesting and useful for many
>reasons, among them: sometimes we can map solutions of a -linear- pde to
>solutions of a -nonlinear- one! (This happens, for example, with the
>sine-Gordon equation mentioned above.)
Am I to infer that work on solitons as a potential explanation for
elementary particles has not ceased, but continues and progresses to
this day?
Unfortunately I only really understood about .1% of the post (hey, I'm
an optimist) but I got the impression that solitons could be generated
in mathematical structures similar to those in common use in theoretical
physics.
I am also intrigued by several statements, one of which I quoted above.
If one can map solutions of a linear pde to a non-linear one then in a
world where you only ever see solutions it would be easy to assume they
were modelled by a linear one and not (which could be the case) the non-
linear one. Given the simplifications allowed by linear anything
compared to non-linear (which tend towards being intractable) one could
be rather badly sidetracked at quite an early point.
--
Oz
This post is worth absolutely nothing and is probably fallacious.
BTOPENWORLD address about to cease. DEMON address no longer in use.
>>Use oz@farmeroz.port995.com (whitelist check on first posting)<<
tessel@tum.bot
Jun22-04, 04:50 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nI wrote:\n\n>> I just said that Lie-Baecklund symmetries can be used to hunt for\n>> Baecklund morphisms. These are mappings from the solution space of one\n>> pde F(x,t,u,u_x..) to that of another pde G(x,t,v,v_x,...), which is\n>> usually of the same order. They are interesting and useful for many\n>> reasons, among them: sometimes we can map solutions of a -linear- pde\n>> to solutions of a -nonlinear- one! (This happens, for example, with\n>> the sine-Gordon equation mentioned above.)\n\nOz had his mind blown :-) and asked:\n\n> Am I to infer that work on solitons as a potential explanation for\n> elementary particles has not ceased, but continues and progresses to\n> this day?\n\nNo! I am by no means an expert on solitons and their applications (just\nhave been doing some reading), so I have no idea if anyone is (or if\nanyone ever was) -seriously- attempting to interpret elementary particle\nphysics in terms of solitons. I do assume, however, that if anything very\nuseful along these lines had been achieved, I would have read about it\nsomewhere by now. One obvious problem with trying to relate what I wrote\n(mostly about the KdV equation) to elementary particle physics is that it\nis not "relativistic", at least not in any sense that I recognize.\n\n> Unfortunately I only really understood about 0.1% of the post (hey, I\'m\n> an optimist) but I got the impression that solitons could be generated\n> in mathematical structures similar to those in common use in theoretical\n> physics.\n\nYes, in fact, while the informal term "soliton" apparently does not have a\nformal definition, everyone agrees that some things count as solitons.\nSuch things appear in the Toda lattice and other models which are\nimportant in theoretical physics.\n\n> I am also intrigued by several statements, one of which I quoted above.\n> If one can\n\n-sometimes-\n\n> map solutions of a linear pde to a non-linear one\n\nProbably I expressed myself badly (writing under difficult circumstances).\nLet me try again:\n\nPdes have solution spaces (a kind of "space" consisting of all solutions\nto the pde in question). Sometimes, for particular pairs of pdes, one\nlinear and one nonlinear, we can find a Baecklund morphism mapping one\nspace into the other. Example: there is such a map from the solution\nspace of the heat equation (linear, with the general solution via\nseparation of variables well known) into the solution space of Burger\'s\nequation (a particular nonlinear pde). This means we can immediately\nobtain an infinite dimensional subspace of the solution space of Burger\'s\nequation (maybe not every solution, though).\n\nContrast the symmetry Ansatz method I sketched, which enables us to obtain\na finite dimensional space of solutions from just one particular solution.\nWhile some trivial particular solutions of Burger\'s equation are evident\nby inspection, this method is limited by the fact that the point symmetry\ngroup of Burger\'s equation is finite dimensional--- clearly, we can obtain\nonly a very small (and highly nonrepresentative) selection of solutions of\nBurger\'s equation by this method. OTH, this method applies even when\nBaecklund morphisms cannot be found.\n\nYet another variation on this theme which I mentioned is that we can look\nfor the "Lie-Baecklund" symmetry group, which is a supergroup of the point\nsymmetry group, but which may be infinite dimensional even if the point\nsymmetry group is finite dimensional. OTH, it is much harder to compute,\nand when we obtain the Lie-Baecklund symmetries, it is harder to use them\nto obtain solutions.\n\nMath and mathematical physics are very hard, even in the simplest\nsituations, so tradeoffs abound.\n\n> then in a world where you only ever see solutions\n\ndoes not compute :-/\n\n> it would be easy to assume they were modelled by a linear one and not\n> (which could be the case) the nonlinear one. Given the simplifications\n> allowed by linear anything compared to non-linear (which tend towards\n> being intractable) one could be rather badly sidetracked at quite an\n> early point.\n\nI guess I don\'t understand the question, but noone is claiming that you\ncan get away with ignoring nonlinearity. Situations where you can obtain\nsolutions of a particular nonlinear PDE from the known "general solution"\nof some linear PDE is rather special; see\n\nauthor = {Bluman, George W., and Kumei, Sukeyuki},\ntitle = {Symmetries and Differential Equations},\nseries = {Applied mathematical sciences},\nvolume =81,\npublisher = {Springer-Verlag},\nyear = {1989}}\n\n"T. Essel" (hiding somewhere in cyberspace)\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>I wrote:
>> I just said that Lie-Baecklund symmetries can be used to hunt for
>> Baecklund morphisms. These are mappings from the solution space of one
>> pde F(x,t,u,u_x..) to that of another pde G(x,t,v,v_x,...), which is
>> usually of the same order. They are interesting and useful for many
>> reasons, among them: sometimes we can map solutions of a -linear- pde
>> to solutions of a -nonlinear- one! (This happens, for example, with
>> the sine-Gordon equation mentioned above.)
Oz had his mind blown :-) and asked:
> Am I to infer that work on solitons as a potential explanation for
> elementary particles has not ceased, but continues and progresses to
> this day?
No! I am by no means an expert on solitons and their applications (just
have been doing some reading), so I have no idea if anyone is (or if
anyone ever was) -seriously- attempting to interpret elementary particle
physics in terms of solitons. I do assume, however, that if anything very
useful along these lines had been achieved, I would have read about it
somewhere by now. One obvious problem with trying to relate what I wrote
(mostly about the KdV equation) to elementary particle physics is that it
is not "relativistic", at least not in any sense that I recognize.
> Unfortunately I only really understood about .1% of the post (hey, I'm
> an optimist) but I got the impression that solitons could be generated
> in mathematical structures similar to those in common use in theoretical
> physics.
Yes, in fact, while the informal term "soliton" apparently does not have a
formal definition, everyone agrees that some things count as solitons.
Such things appear in the Toda lattice and other models which are
important in theoretical physics.
> I am also intrigued by several statements, one of which I quoted above.
> If one can
-sometimes-
> map solutions of a linear pde to a non-linear one
Probably I expressed myself badly (writing under difficult circumstances).
Let me try again:
Pdes have solution spaces (a kind of "space" consisting of all solutions
to the pde in question). Sometimes, for particular pairs of pdes, one
linear and one nonlinear, we can find a Baecklund morphism mapping one
space into the other. Example: there is such a map from the solution
space of the heat equation (linear, with the general solution via
separation of variables well known) into the solution space of Burger's
equation (a particular nonlinear pde). This means we can immediately
obtain an infinite dimensional subspace of the solution space of Burger's
equation (maybe not every solution, though).
Contrast the symmetry Ansatz method I sketched, which enables us to obtain
a finite dimensional space of solutions from just one particular solution.
While some trivial particular solutions of Burger's equation are evident
by inspection, this method is limited by the fact that the point symmetry
group of Burger's equation is finite dimensional--- clearly, we can obtain
only a very small (and highly nonrepresentative) selection of solutions of
Burger's equation by this method. OTH, this method applies even when
Baecklund morphisms cannot be found.
Yet another variation on this theme which I mentioned is that we can look
for the "Lie-Baecklund" symmetry group, which is a supergroup of the point
symmetry group, but which may be infinite dimensional even if the point
symmetry group is finite dimensional. OTH, it is much harder to compute,
and when we obtain the Lie-Baecklund symmetries, it is harder to use them
to obtain solutions.
Math and mathematical physics are very hard, even in the simplest
situations, so tradeoffs abound.
> then in a world where you only ever see solutions
does not compute :-/
> it would be easy to assume they were modelled by a linear one and not
> (which could be the case) the nonlinear one. Given the simplifications
> allowed by linear anything compared to non-linear (which tend towards
> being intractable) one could be rather badly sidetracked at quite an
> early point.
I guess I don't understand the question, but noone is claiming that you
can get away with ignoring nonlinearity. Situations where you can obtain
solutions of a particular nonlinear PDE from the known "general solution"
of some linear PDE is rather special; see
author = {Bluman, George W., and Kumei, Sukeyuki},
title = {Symmetries and Differential Equations},
series = {Applied mathematical sciences},
volume =81,
publisher = {Springer-Verlag},
year = {1989}}
"T. Essel" (hiding somewhere in cyberspace)
Gerard Westendorp
Jun28-04, 12:10 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\ntessel@tum.bot wrote:\n\n[.. cool summary about solitons..]\n\n\n> This turns out to be closely related to the existence of an infinite\n> hierarchy of conservation laws. As you probably know, one of Poincare\'s\n> greatest discoveries was the fact that, in some sense, "most" Hamiltonian\n> systems are not solvable in closed form (e.g. the -general- three body\n> problem of Newtonian gravitation). The KdV, OTH, admits infinitely many\n> "integrals of motion" (corresponding to the conserved quantities mentioned\n> above). This is characteristic of a "completely integrable Hamiltonian\n> system". (See again my remarks above concerning n-soliton solutions of\n> the KdV and conserved quantities.)\n\n\nHow would this work out for the linear wave equation?\n\nMy guess:\nThe infinite conserved quantities are just the energies of the\nlinearly independent eigenmodes. These can be rewritten as\nthings like total energy, total momentum, total charge, etc.\n\nMaybe "completely integrable Hamiltonian systems" are a bit\nlike "warped" linear systems, whereas the other ones are not.\n\nActually, I do not quite understand what it means physically to\nhave a non- completely integrable system. Chaos?\n\nGerard\n\n\n\n\nGerard\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>tessel@tum.bot wrote:
[.. cool summary about solitons..]
> This turns out to be closely related to the existence of an infinite
> hierarchy of conservation laws. As you probably know, one of Poincare's
> greatest discoveries was the fact that, in some sense, "most" Hamiltonian
> systems are not solvable in closed form (e.g. the -general- three body
> problem of Newtonian gravitation). The KdV, OTH, admits infinitely many
> "integrals of motion" (corresponding to the conserved quantities mentioned
> above). This is characteristic of a "completely integrable Hamiltonian
> system". (See again my remarks above concerning n-soliton solutions of
> the KdV and conserved quantities.)
How would this work out for the linear wave equation?
My guess:
The infinite conserved quantities are just the energies of the
linearly independent eigenmodes. These can be rewritten as
things like total energy, total momentum, total charge, etc.
Maybe "completely integrable Hamiltonian systems" are a bit
like "warped" linear systems, whereas the other ones are not.
Actually, I do not quite understand what it means physically to
have a non- completely integrable system. Chaos?
Gerard
Gerard
Gerard Westendorp
Jun29-04, 04:44 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Oz wrote:\n\n[..]\n\n\n> Am I to infer that work on solitons as a potential explanation for\n> elementary particles has not ceased, but continues and progresses to\n> this day?\n>\n\n\nOne thing thing a soliton model of lets say the electron/muon/tau-on\nwould have to explain is the 3 different masses. This might be\nsomething a soliton model could do: they predict fixed amplitudes.\n\nBut how could a soliton model explain that all 3 particles have\nthe same charge, and the same angular momentum?\n\nGerard\n\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Oz wrote:
[..]
> Am I to infer that work on solitons as a potential explanation for
> elementary particles has not ceased, but continues and progresses to
> this day?
>
One thing thing a soliton model of lets say the electron/muon/\tau-on
would have to explain is the 3 different masses. This might be
something a soliton model could do: they predict fixed amplitudes.
But how could a soliton model explain that all 3 particles have
the same charge, and the same angular momentum?
Gerard
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Gerard Westendorp <westy31@xs4all.nl> writes\n>Oz wrote:\n>\n>[..]\n>\n>\n>> Am I to infer that work on solitons as a potential explanation for\n>> elementary particles has not ceased, but continues and progresses to\n>> this day?\n>>\n>\n>\n>One thing thing a soliton model of lets say the electron/muon/tau-on\n>would have to explain is the 3 different masses. This might be\n>something a soliton model could do: they predict fixed amplitudes.\n>\n>But how could a soliton model explain that all 3 particles have\n>the same charge, and the same angular momentum?\n\nClearly I am not the one to know this.\n\nHowever it doesn\'t seem implausible to me that a wave in a non-linear\nenvironment could have more than one solution depending on amplitude.\n\nIndeed, in the days of (expensive) valves, which were not at all linear,\nit was quite common to find (in home-made home-designed audio circuits)\n\'ringing\' (that is self-oscillation) at several different levels that\nwere stable and repeatable. Two (one low level and one very high level)\nwas very common indeed, and three or four came along from time to time.\nOf course redesign removed these by judicious alteration of circuit\nparameters (and sometimes speaker location).\n\nI don\'t remember if these had the same frequency, but from memory it was\nclose as one typically didn\'t adjust the scope timebase, but merely\nchanged sensitivity (and sometimes not even that).\n\nHmmm. This sort of concept could lend itself to species mixing, I would\nhave thought. Certainly superposition becomes expected.\n\n--\nOz\nThis post is worth absolutely nothing and is probably fallacious.\n\nBTOPENWORLD address about to cease. DEMON address no longer in use.\n>>Use oz@farmeroz.port995.com (whitelist check on first posting)<<\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Gerard Westendorp <westy31@xs4all.nl> writes
>Oz wrote:
>
>[..]
>
>
>> Am I to infer that work on solitons as a potential explanation for
>> elementary particles has not ceased, but continues and progresses to
>> this day?
>>
>
>
>One thing thing a soliton model of lets say the electron/muon/\tau-on
>would have to explain is the 3 different masses. This might be
>something a soliton model could do: they predict fixed amplitudes.
>
>But how could a soliton model explain that all 3 particles have
>the same charge, and the same angular momentum?
Clearly I am not the one to know this.
However it doesn't seem implausible to me that a wave in a non-linear
environment could have more than one solution depending on amplitude.
Indeed, in the days of (expensive) valves, which were not at all linear,
it was quite common to find (in home-made home-designed audio circuits)
'ringing' (that is self-oscillation) at several different levels that
were stable and repeatable. Two (one low level and one very high level)
was very common indeed, and three or four came along from time to time.
Of course redesign removed these by judicious alteration of circuit
parameters (and sometimes speaker location).
I don't remember if these had the same frequency, but from memory it was
close as one typically didn't adjust the scope timebase, but merely
changed sensitivity (and sometimes not even that).
Hmmm. This sort of concept could lend itself to species mixing, I would
have thought. Certainly superposition becomes expected.
--
Oz
This post is worth absolutely nothing and is probably fallacious.
BTOPENWORLD address about to cease. DEMON address no longer in use.
>>Use oz@farmeroz.port995.com (whitelist check on first posting)<<
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>tessel@tum.bot writes\n>\n>\n>Oz had his mind blown :-)\n\nA rather common experience here.\nThe universe behaves in a hugely satisfying mind-blowing manner.\n\n>and asked:\n>\n>> Am I to infer that work on solitons as a potential explanation for\n>> elementary particles has not ceased, but continues and progresses to\n>> this day?\n>\n>No! I am by no means an expert on solitons and their applications (just\n>have been doing some reading), so I have no idea if anyone is (or if\n>anyone ever was) -seriously- attempting to interpret elementary particle\n>physics in terms of solitons.\n\nOh...\nPity...\nI always liked the idea conceptually.\nI understood that the maths was, er, \'challenging\'.\n\nPhilosophically I find them attractive too. In general we seem to find\nphysics starts off linear (eg newton gravity and mechanics) but needs to\nbe refined into increasingly non-linear models (SR, GR) to model more\nextreme situations. In particular we see less of "A affects B, but B\ndoesn\'t affect A" and more and more "A affects B, and B affects A".\nThese tend to be in a non-linear manner that leads to a certain\nmathematical complexity.\n\nSo the idea that a particle alters the space its in, and space alters\nthe particle that\'s in it is highly attractive as a model. Solitons\noffer one neat explanation to particles in that only a limited number of\npossible stable modes might be expected.\n\nI also note that EM still remains linear. At extremes I would expect it\nto show signs of non-linearity too.\n\n>I do assume, however, that if anything very\n>useful along these lines had been achieved, I would have read about it\n>somewhere by now.\n\nYes.\n\n>One obvious problem with trying to relate what I wrote\n>(mostly about the KdV equation) to elementary particle physics is that it\n>is not "relativistic", at least not in any sense that I recognize.\n\nThat\'s a big problem. On the other hand to first order particles are at\nrest with respect to themselves so it might not be quite the problem it\nappears. After all if you (say) obtained a solution for an electronlike\nsoliton with three solutions that bore any sort of resemblance to what\nwe see, I think that would be a triumph.\n\nI can\'t see, though, how to make up a plausible non-linearity. You would\nperhaps be looking for something like gravity that was strong at short\ndistances and weak at long distances. That is, behaved like a photon\n(EM) at long distances (but weak) and the weak at short distances (but\nstrong). Hmm, didn\'t someone once say that the weak force is in fact a\nvery strong force, but just very short distance? Anyway, I am rambling.\n\n>> Unfortunately I only really understood about 0.1% of the post (hey, I\'m\n>> an optimist) but I got the impression that solitons could be generated\n>> in mathematical structures similar to those in common use in theoretical\n>> physics.\n>\n>Yes, in fact, while the informal term "soliton" apparently does not have a\n>formal definition, everyone agrees that some things count as solitons.\n\nHmm, suggests that study of solitons has been fraught with difficulties.\n\n>Such things appear in the Toda lattice and other models which are\n>important in theoretical physics.\n\nGood.\n\n>> I am also intrigued by several statements, one of which I quoted above.\n>> If one can\n>\n>-sometimes-\n>\n>> map solutions of a linear pde to a non-linear one\n>\n>Probably I expressed myself badly (writing under difficult circumstances).\n>Let me try again:\n>\n>Pdes have solution spaces (a kind of "space" consisting of all solutions\n>to the pde in question). Sometimes, for particular pairs of pdes, one\n>linear and one nonlinear, we can find a Baecklund morphism mapping one\n>space into the other. Example: there is such a map from the solution\n>space of the heat equation (linear, with the general solution via\n>separation of variables well known) into the solution space of Burger\'s\n>equation (a particular nonlinear pde). This means we can immediately\n>obtain an infinite dimensional subspace of the solution space of Burger\'s\n>equation (maybe not every solution, though).\n\nThat seems very similar, certainly in concept.\n\n>Contrast the symmetry Ansatz method I sketched, which enables us to obtain\n>a finite dimensional space of solutions from just one particular solution.\n>While some trivial particular solutions of Burger\'s equation are evident\n>by inspection, this method is limited by the fact that the point symmetry\n>group of Burger\'s equation is finite dimensional--- clearly, we can obtain\n>only a very small (and highly nonrepresentative) selection of solutions of\n>Burger\'s equation by this method. OTH, this method applies even when\n>Baecklund morphisms cannot be found.\n\nOf course I have not the faintest idea of the structures you are\ndescribing, but no matter. My question is whether the mapping is of some\nsolutions of the non-linear pde to the linear one. That is we can get\nsome of the non-linear solutions but not all of them. Or is it vice\nversa? Hmm I know you are going to say \'it depends\'.....\n\n>Yet another variation on this theme which I mentioned is that we can look\n>for the "Lie-Baecklund" symmetry group, which is a supergroup of the point\n>symmetry group, but which may be infinite dimensional even if the point\n>symmetry group is finite dimensional. OTH, it is much harder to compute,\n>and when we obtain the Lie-Baecklund symmetries, it is harder to use them\n>to obtain solutions.\n>\n>Math and mathematical physics are very hard, even in the simplest\n>situations, so tradeoffs abound.\n\nYes. And the simplicity of linearity must be temptingly attractive.\n\n>> then in a world where you only ever see solutions\n>\n>does not compute :-/\n\nOh. What I meant was (and I hope this is right) that typically, given a\nmathematical structure modelling physics, (selected) solutions to the\nequations of the model are typically what we observe. That is, for\nexample, we often set up a pde to describe movement of a body\n(gravitationally for example), apply boundary conditions and the\nsolution gives us the observed path of the body. I will accept that I am\nlikely being naive here.\n\n>> it would be easy to assume they were modelled by a linear one and not\n>> (which could be the case) the nonlinear one. Given the simplifications\n>> allowed by linear anything compared to non-linear (which tend towards\n>> being intractable) one could be rather badly sidetracked at quite an\n>> early point.\n>\n>I guess I don\'t understand the question, but noone is claiming that you\n>can get away with ignoring nonlinearity. Situations where you can obtain\n>solutions of a particular nonlinear PDE from the known "general solution"\n>of some linear PDE is rather special; see\n\n<snip>\n\nIndeed. But in a situation where you don\'t even know the form of the\nnon-linear equation and are grasping blindly for a mathematical\ndescription of experimental results, linear (that is more manageable and\nuseful) descriptions are likely to be the first port of call. Then these\nget modified to tweak a linear system trying to match a non-linear one.\nThen someone hits on the non-linear one....\n\nWhich, in general, you can\'t solve .....\n\n>"T. Essel" (hiding somewhere in cyberspace)\n\nHmmm. And concealing his/her tracks well....\n\n--\nOz\nThis post is worth absolutely nothing and is probably fallacious.\n\nBTOPENWORLD address about to cease. DEMON address no longer in use.\n>>Use oz@farmeroz.port995.com (whitelist check on first posting)<<\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>tessel@tum.bot writes
>
>
>Oz had his mind blown :-)
A rather common experience here.
The universe behaves in a hugely satisfying mind-blowing manner.
>and asked:
>
>> Am I to infer that work on solitons as a potential explanation for
>> elementary particles has not ceased, but continues and progresses to
>> this day?
>
>No! I am by no means an expert on solitons and their applications (just
>have been doing some reading), so I have no idea if anyone is (or if
>anyone ever was) -seriously- attempting to interpret elementary particle
>physics in terms of solitons.
Oh...
Pity...
I always liked the idea conceptually.
I understood that the maths was, er, 'challenging'.
Philosophically I find them attractive too. In general we seem to find
physics starts off linear (eg newton gravity and mechanics) but needs to
be refined into increasingly non-linear models (SR, GR) to model more
extreme situations. In particular we see less of "A affects B, but B
doesn't affect A" and more and more "A affects B, and B affects A".
These tend to be in a non-linear manner that leads to a certain
mathematical complexity.
So the idea that a particle alters the space its in, and space alters
the particle that's in it is highly attractive as a model. Solitons
offer one neat explanation to particles in that only a limited number of
possible stable modes might be expected.
I also note that EM still remains linear. At extremes I would expect it
to show signs of non-linearity too.
>I do assume, however, that if anything very
>useful along these lines had been achieved, I would have read about it
>somewhere by now.
Yes.
>One obvious problem with trying to relate what I wrote
>(mostly about the KdV equation) to elementary particle physics is that it
>is not "relativistic", at least not in any sense that I recognize.
That's a big problem. On the other hand to first order particles are at
rest with respect to themselves so it might not be quite the problem it
appears. After all if you (say) obtained a solution for an electronlike
soliton with three solutions that bore any sort of resemblance to what
we see, I think that would be a triumph.
I can't see, though, how to make up a plausible non-linearity. You would
perhaps be looking for something like gravity that was strong at short
distances and weak at long distances. That is, behaved like a photon
(EM) at long distances (but weak) and the weak at short distances (but
strong). Hmm, didn't someone once say that the weak force is in fact a
very strong force, but just very short distance? Anyway, I am rambling.
>> Unfortunately I only really understood about .1% of the post (hey, I'm
>> an optimist) but I got the impression that solitons could be generated
>> in mathematical structures similar to those in common use in theoretical
>> physics.
>
>Yes, in fact, while the informal term "soliton" apparently does not have a
>formal definition, everyone agrees that some things count as solitons.
Hmm, suggests that study of solitons has been fraught with difficulties.
>Such things appear in the Toda lattice and other models which are
>important in theoretical physics.
Good.
>> I am also intrigued by several statements, one of which I quoted above.
>> If one can
>
>-sometimes-
>
>> map solutions of a linear pde to a non-linear one
>
>Probably I expressed myself badly (writing under difficult circumstances).
>Let me try again:
>
>Pdes have solution spaces (a kind of "space" consisting of all solutions
>to the pde in question). Sometimes, for particular pairs of pdes, one
>linear and one nonlinear, we can find a Baecklund morphism mapping one
>space into the other. Example: there is such a map from the solution
>space of the heat equation (linear, with the general solution via
>separation of variables well known) into the solution space of Burger's
>equation (a particular nonlinear pde). This means we can immediately
>obtain an infinite dimensional subspace of the solution space of Burger's
>equation (maybe not every solution, though).
That seems very similar, certainly in concept.
>Contrast the symmetry Ansatz method I sketched, which enables us to obtain
>a finite dimensional space of solutions from just one particular solution.
>While some trivial particular solutions of Burger's equation are evident
>by inspection, this method is limited by the fact that the point symmetry
>group of Burger's equation is finite dimensional--- clearly, we can obtain
>only a very small (and highly nonrepresentative) selection of solutions of
>Burger's equation by this method. OTH, this method applies even when
>Baecklund morphisms cannot be found.
Of course I have not the faintest idea of the structures you are
describing, but no matter. My question is whether the mapping is of some
solutions of the non-linear pde to the linear one. That is we can get
some of the non-linear solutions but not all of them. Or is it vice
versa? Hmm I know you are going to say 'it depends'.....
>Yet another variation on this theme which I mentioned is that we can look
>for the "Lie-Baecklund" symmetry group, which is a supergroup of the point
>symmetry group, but which may be infinite dimensional even if the point
>symmetry group is finite dimensional. OTH, it is much harder to compute,
>and when we obtain the Lie-Baecklund symmetries, it is harder to use them
>to obtain solutions.
>
>Math and mathematical physics are very hard, even in the simplest
>situations, so tradeoffs abound.
Yes. And the simplicity of linearity must be temptingly attractive.
>> then in a world where you only ever see solutions
>
>does not compute :-/
Oh. What I meant was (and I hope this is right) that typically, given a
mathematical structure modelling physics, (selected) solutions to the
equations of the model are typically what we observe. That is, for
example, we often set up a pde to describe movement of a body
(gravitationally for example), apply boundary conditions and the
solution gives us the observed path of the body. I will accept that I am
likely being naive here.
>> it would be easy to assume they were modelled by a linear one and not
>> (which could be the case) the nonlinear one. Given the simplifications
>> allowed by linear anything compared to non-linear (which tend towards
>> being intractable) one could be rather badly sidetracked at quite an
>> early point.
>
>I guess I don't understand the question, but noone is claiming that you
>can get away with ignoring nonlinearity. Situations where you can obtain
>solutions of a particular nonlinear PDE from the known "general solution"
>of some linear PDE is rather special; see
<snip>
Indeed. But in a situation where you don't even know the form of the
non-linear equation and are grasping blindly for a mathematical
description of experimental results, linear (that is more manageable and
useful) descriptions are likely to be the first port of call. Then these
get modified to tweak a linear system trying to match a non-linear one.
Then someone hits on the non-linear one....
Which, in general, you can't solve .....
>"T. Essel" (hiding somewhere in cyberspace)
Hmmm. And concealing his/her tracks well....
--
Oz
This post is worth absolutely nothing and is probably fallacious.
BTOPENWORLD address about to cease. DEMON address no longer in use.
>>Use oz@farmeroz.port995.com (whitelist check on first posting)<<
tessel@tum.bot
Jul2-04, 04:32 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Wed, 30 Jun 2004, Oz wrote:\n\n> tessel@tum.bot writes\n\n> >I have no idea if anyone is (or if anyone ever was) -seriously-\n> >attempting to interpret elementary particle physics in terms of\n> >solitons.\n>\n> Oh...\n> Pity...\n> I always liked the idea conceptually.\n\nI see I missed an opportunity to say something else which is (?)\nmindblowing.\n\nMiura found a Baecklund transformation relating the KdV equation and what\nI was calling the Miura equation (sometimes called mKdV equation). Miura\nfurther observed that this leads directly to a connection with the\nSchroedinger equation! This is basically how the inverse scattering\ntransformation method of solving the KdV was discovered, in fact.\n(Actually, for finding "asymptotically flat solutions of the KdV"--- which\nis just what we want when applying the IST to the problem of finding\n"asympotically flat solutions of the EFE"; hence "gravitational\nsolitons".)\n\nBTW, don\'t confuse "soliton" with "solitary wave" (which just means a\nsingle wavecrest progagating as a traveling wave, i.e. a much more general\nconcept). The best short definition of "soliton" is probably something\nlike this: "solitary wave solution to a PDE solvable by the IST".\n\nEven before solitons came along, people had already asked and answered the\nquestion: can I determine the nature of the potential in the Schroedinger\nequation if I know enough about how an unknown potential\nscatters/transmits plane waves? The answer is yes. Combined with Miura\'s\namazing insights, this leads to the IST method of solving KdV. Lax and\nothers then generalized this to solve a large class of equations by\nsomething which still deserved to be called "the IST"; solitary wave\nsolutions of these are solitons.\n\nHere\'s the mindblowing bit: you know that Schroedinger\'s equation is\ninvolved with various mystical utterances about electrons "having both a\nwave and a particle nature". But recall the behavior of the n-soliton\nsolutions of the KdV. We saw that the solitons are particle-like in some\nrespects; they are localized (if you squint, almost pointlike), they can\ncollide. But they are solutions to a wave equation, so they are also\nwave-like in some respects.\n\nHmm... actually, as I stated it, this seems rather mealy-mouthed, not very\nimpressive at all. To get a better impression of why the idea ever arose\nthat solitons might have something profound to say about particle/wave\nduality in the Schroedinger equation, one should learn about the\nconnection with "mechanical models" like the rubber band model for the\nsine-Gordon equation I mentioned. This is discussed in--- oh!\n\nI just realized the book I am about to mention is -perfect- for -you-, Oz!\nIt is:\n\nauthor = {E. Atlee Jackson},\ntitle = {Perspectives of Nonlinear Dynamics},\nnote = {Two Volumes},\npublisher = {Cambridge University Press},\nyear = 1991}\n\nThis is very readable and discusses -all- of the above and much much more,\nbut better yet, it has such great illustrations that you, Oz, really have\na good chance of learning quite a bit even if you skip the equations. At\nleast, I hope so. It is a cheap paperback, and I\'m pretty sure it\'s still\nin print. The stuff you want is in volume 2, but wouldn\'t you know it,\nyou probably need volume 1 to understand volume 2. But nonlinear dynamics\nin general is -great- stuff, really fun, and Jackson does an amazing job\nof explaining so much using mostly English and pictures. Also, the author\nhad something to do with the development of the "mechanical model" aspect\nof soliton theory, so his discussion of that is particularly good!\n\nAnother omission--- did I mention the fish tank experiment? If you have a\nflat bottomed aquarium with flat glass walls (one which is uninhabited by\nany fish, of course!), fill it with a half inch of water. Possibly adding\nfood coloring to the water will help make the surface more visible. Get a\nbrick or something of that general shape which will fit nicely at one end\nof the tank. Hold it just over the water at one end and drop so it falls\nsuddenly but not violently. You shoud be able to reliably generate a\nsoliton wave this way. With a long straight flat bottomed shallow water\nchannel and several bricks, you can, with more effort, generate counter\npropagating solitons which collide and interact.\n\nHmmm... Jackson suggests a variation of this employing a public swimming\npool, but I don\'t know if you have one of those on the farm. The key is\n-shallow- water and a channel which is not too short. A long flat\nbottomed feeding trough might work.\n\nAlso, if you can find a shallow square tank (maybe a window removed from\nits frame and set flat?), replacing the brick with a long heavy dowel cut\nto find one end of the window, you can generate straight-sided ripples.\nAdd another rod and you can try to get these "traveling ridges" to\ninteract like solitons (there are higher dimensional versions of the KdV\nwhich model these ridge-like solitons). Try to create a Y shaped\ninteraction of two of these!\n\n> Philosophically I find them attractive too. In general we seem to find\n> physics starts off linear (eg newton gravity and mechanics)\n\nYrmm... well, when I said "nonlinear" I really meant something more like\nthis:\n\n1. solitary wavecrest with nonlinear amplitude-dependent velocity,\n\n2. dispersion,\n\n3. dissipation.\n\nNow contrast these two notions:\n\nPhenomenon Examples\n\n(1,2) solitons KdV & Miura equations\n\n(1,3) shocks Burger\'s equation\n\n> but needs to be refined into increasingly non-linear models (SR, GR) to\n> model more extreme situations.\n\nI don\'t know what you mean by saying str is "nonlinear".\n\nIf we accept that "the principle of relativity" states that "fundamental\nphysical laws are Lorentz invariant", then since the Lorentz group is a\ngroup of affine transformations. Affine transformations are obtained by\ncombining translations, which can move the origin, with linear\ntransformations, which cannot, so they are "linear" in the sense that they\ntake lines to lines, planes to planes.\n\nOr again, in historical context, str gives the correct geometric setting\nfor Maxwell\'s theory of EM, which we call "flat spacetime". Gtr\ngeneralizes the good things which follow to curved spacetimes, but\nincorporates the "principle of equivalence" in a very simple and\nattractive way, which leads to a theory of "gravity as geometry". Very\nroughly speaking.\n\n> In particular we see less of "A affects B, but B doesn\'t affect A" and\n> more and more "A affects B, and B affects A". These tend to be in a\n> non-linear manner that leads to a certain mathematical complexity.\n\nMaybe you are thinking of Feynman\'s intuitive explanation (in his\nLectures) of why Maxwell\'s equations allow the propagation of EM waves?\nIf so, this is str, and Maxwell\'s equations are linear in the sense that\ntaking any linear combination f = c1 f1 + c2 f2 of two old solutions f1,\nf2 gives a new solution.\n\n> So the idea that a particle alters the space its in, and space alters\n> the particle that\'s in it is highly attractive as a model.\n\nI think you might be confusing "a theory which admits feedback" with "a\nnonlinear theory". But feedback of a kind can be linear; see Feynman\'s\ndiscussion just cited.\n\n> Solitons offer one neat explanation to particles in that only a limited\n> number of possible stable modes might be expected.\n\nWell, I and Gerard mentioned some obvious problems. Another problem is\nthat the velocity dependence of solitary wave crests in solutions to the\nKdV is much more restrictive than particles in mechanics. The real\nmotivation is more abstract and has to do with the IST as sketched above.\n\n> I also note that EM still remains linear.\n\nYes.\n\n> At extremes I would expect it to show signs of non-linearity too.\n\nIn a sense, in gtr it is linear locally, but in the large, yes, the EM\nfield energy gravitates and like in gtr all gravitation is nonlinear.\nBut for weak gravity the nonlinear effects are too small to see, which is\nwhy we got away with Newtonian theory for so long. And human generated EM\nfields, or even astrophysical EM fields, probably cannot generate\nsufficient densities of field energy to significantly alter ambient\ngravitational fields. Too bad, since theoretical solutions like the\nMelvin electrovacuum (a magnetic field flux tube held together by its own\nself-gravitation) are such fun!\n\nHowever, on the ArXiv you can find some papers speculating (in more or\nless precise mathematical terms, of course!) about "nonlinear EM". Of\ncourse any such theory should reduce to Maxwell for weak fields. (The\ntheories I am thinking of are not neccessarily intended as "unifications"\nof EM with anything, BTW, just as speculations about the possibility that\nif we could generate -really- intense EM fields, we\'d find something less\nfamiliar to theorists than Einstein-Maxwell.)\n\n> >One obvious problem with trying to relate what I wrote (mostly about\n> >the KdV equation) to elementary particle physics is that it is not\n> >"relativistic", at least not in any sense that I recognize.\n>\n> That\'s a big problem. On the other hand to first order particles are at\n> rest with respect to themselves so it might not be quite the problem it\n> appears. After all if you (say) obtained a solution for an electronlike\n> soliton with three solutions that bore any sort of resemblance to what\n> we see, I think that would be a triumph.\n\nUhm...\n\n> I can\'t see, though, how to make up a plausible non-linearity. You would\n> perhaps be looking for something like gravity that was strong at short\n> distances and weak at long distances. That is, behaved like a photon\n> (EM) at long distances (but weak) and the weak at short distances (but\n> strong).\n\nI understand that both Feynman and Seinfeld have spoken at Cornell, but I\nthink this is from the latter\'s presentation :-/\n\n> >Yes, in fact, while the informal term "soliton" apparently does not have a\n> >formal definition, everyone agrees that some things count as solitons.\n>\n> Hmm, suggests that study of solitons has been fraught with difficulties.\n\nIt suggests that "soliton theory" is a still a work in progress, but the\n"theory" even in its present form has been very successful.\n\n> >Contrast the symmetry Ansatz method I sketched, which enables us to\n> >obtain a finite dimensional space of solutions from just one particular\n> >solution. While some trivial particular solutions of Burger\'s equation\n> >are evident by inspection, this method is limited by the fact that the\n> >point symmetry group of Burger\'s equation is finite dimensional---\n> >clearly, we can obtain only a very small (and highly nonrepresentative)\n> >selection of solutions of Burger\'s equation by this method. OTH, this\n> >method applies even when Baecklund morphisms cannot be found.\n>\n> Of course I have not the faintest idea of the structures you are\n> describing, but no matter.\n\nSee again what I said about how the IST was discovered.\n\n> My question is whether the mapping is of some solutions of the\n> non-linear pde to the linear one.\n\nOther way around: from a solution of the (linear) heat equation, we obtain\na solution of the (nonlinear) Burger\'s equation.\n\nSee above for another point: among other things, I was actually trying to\ncontrast solitons (conservative, dispersive) with shocks (nonconserative,\ndissipative). Two quite different nonlinear phenomena, with two quite\ndifferent canonical examples (KdV, Burger\'s respectively).\n\n> That is we can get some of the non-linear solutions but not all of them.\n> Or is it vice versa? Hmm I know you are going to say \'it depends\'.....\n\nI was trying to suggest that we might not obtain -all- solutions of\nBurger\'s equation in this way. I.e., our Baecklund morphism (the once\nassociated with the Hopf-Cole transformation) from the solution space of\nthe heat equation into the solution space of Burgers\' equatation might not\nbe onto.\n\n> >Yet another variation on this theme which I mentioned is that we can look\n> >for the "Lie-Baecklund" symmetry group, which is a supergroup of the point\n> >symmetry group, but which may be infinite dimensional even if the point\n> >symmetry group is finite dimensional. OTH, it is much harder to compute,\n> >and when we obtain the Lie-Baecklund symmetries, it is harder to use them\n> >to obtain solutions.\n> >\n> >Math and mathematical physics are very hard, even in the simplest\n> >situations, so tradeoffs abound.\n>\n> Yes. And the simplicity of linearity must be temptingly attractive.\n\nWell, simplicity -is- very attractive, but what really tempts here is the\nfact that the general solution of the heat equation is known from\nSturm-Liouville and Fourier, so there we obtain a huge class of solutions\nof Burger\'s equation (possibly not all solutions of Burger\'s equation).\n\n> typically, given a mathematical structure modelling physics, (selected)\n> solutions to the equations of the model are typically what we observe.\n> That is, for example, we often set up a pde to describe movement of a\n> body (gravitationally for example), apply boundary conditions\n\nsee Sturm-Liouville above.\n\n> and the solution gives us the observed path of the body. I will accept\n> that I am likely being naive here.\n\nMaybe you are thinking of Rene Thom\'s notion of "structural stability"?\n\n> >noone is claiming that you can get away with ignoring nonlinearity.\n> >Situations where you can obtain solutions of a particular nonlinear PDE\n> >from the known "general solution" of some linear PDE is rather special;\n> >see\n>\n> Indeed. But in a situation where you don\'t even know the form of the\n> non-linear equation and are grasping blindly for a mathematical\n> description of experimental results, linear (that is more manageable and\n> useful) descriptions are likely to be the first port of call.\n\nPossibly, yes, although one huge lesson of modern dynamical systems theory\nis that we now know lots of ways to handle nonlinear models too, and much\nbetter insight into when linear models probably have no hope of cutting\nthe mustard.\n\n> Then these get modified to tweak a linear system trying to match a\n> non-linear one. Then someone hits on the non-linear one....\n>\n> Which, in general, you can\'t solve .....\n\nMaybe you are thinking of the fact that a possibly nonlinear gravitation\ntheory (e.g gtr) must have a Newtonian limit (linear)? If so, the most\ninteresting examples in say the book by Jackson are arguably completely\nnew phenomena which did -not- arise from trying to extend some previously\nknown linear theory. For example, solitons are inherently nonlinear; I\nam pretty sure that there is nothing like a weak-field limit of a soliton\nwhich can be treated by a linear theory.\n\n"T. Essel" (annihilistically contemplating crop circle creation)\n\nNote: attempts to locate this bot too precisely will result in large\nenergy releases in someone\'s field :-/\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Wed, 30 Jun 2004, Oz wrote:
> tessel@tum.bot writes
> >I have no idea if anyone is (or if anyone ever was) -seriously-
> >attempting to interpret elementary particle physics in terms of
> >solitons.
>
> Oh...
> Pity...
> I always liked the idea conceptually.
I see I missed an opportunity to say something else which is (?)
mindblowing.
Miura found a Baecklund transformation relating the KdV equation and what
I was calling the Miura equation (sometimes called mKdV equation). Miura
further observed that this leads directly to a connection with the
Schroedinger equation! This is basically how the inverse scattering
transformation method of solving the KdV was discovered, in fact.
(Actually, for finding "asymptotically flat solutions of the KdV"--- which
is just what we want when applying the IST to the problem of finding
"asympotically flat solutions of the EFE"; hence "gravitational
solitons".)
BTW, don't confuse "soliton" with "solitary wave" (which just means a
single wavecrest progagating as a traveling wave, i.e. a much more general
concept). The best short definition of "soliton" is probably something
like this: "solitary wave solution to a PDE solvable by the IST".
Even before solitons came along, people had already asked and answered the
question: can I determine the nature of the potential in the Schroedinger
equation if I know enough about how an unknown potential
scatters/transmits plane waves? The answer is yes. Combined with Miura's
amazing insights, this leads to the IST method of solving KdV. Lax and
others then generalized this to solve a large class of equations by
something which still deserved to be called "the IST"; solitary wave
solutions of these are solitons.
Here's the mindblowing bit: you know that Schroedinger's equation is
involved with various mystical utterances about electrons "having both a
wave and a particle nature". But recall the behavior of the n-soliton
solutions of the KdV. We saw that the solitons are particle-like in some
respects; they are localized (if you squint, almost pointlike), they can
collide. But they are solutions to a wave equation, so they are also
wave-like in some respects.
Hmm... actually, as I stated it, this seems rather mealy-mouthed, not very
impressive at all. To get a better impression of why the idea ever arose
that solitons might have something profound to say about particle/wave
duality in the Schroedinger equation, one should learn about the
connection with "mechanical models" like the rubber band model for the
sine-Gordon equation I mentioned. This is discussed in--- oh!
I just realized the book I am about to mention is -perfect- for -you-, Oz!
It is:
author = {E. Atlee Jackson},
title = {Perspectives of Nonlinear Dynamics},
note = {Two Volumes},
publisher = {Cambridge University Press},
year = 1991}
This is very readable and discusses -all- of the above and much much more,
but better yet, it has such great illustrations that you, Oz, really have
a good chance of learning quite a bit even if you skip the equations. At
least, I hope so. It is a cheap paperback, and I'm pretty sure it's still
in print. The stuff you want is in volume 2, but wouldn't you know it,
you probably need volume 1 to understand volume 2. But nonlinear dynamics
in general is -great- stuff, really fun, and Jackson does an amazing job
of explaining so much using mostly English and pictures. Also, the author
had something to do with the development of the "mechanical model" aspect
of soliton theory, so his discussion of that is particularly good!
Another omission--- did I mention the fish tank experiment? If you have a
flat bottomed aquarium with flat glass walls (one which is uninhabited by
any fish, of course!), fill it with a half inch of water. Possibly adding
food coloring to the water will help make the surface more visible. Get a
brick or something of that general shape which will fit nicely at one end
of the tank. Hold it just over the water at one end and drop so it falls
suddenly but not violently. You shoud be able to reliably generate a
soliton wave this way. With a long straight flat bottomed shallow water
channel and several bricks, you can, with more effort, generate counter
propagating solitons which collide and interact.
Hmmm... Jackson suggests a variation of this employing a public swimming
pool, but I don't know if you have one of those on the farm. The key is
-shallow- water and a channel which is not too short. A long flat
bottomed feeding trough might work.
Also, if you can find a shallow square tank (maybe a window removed from
its frame and set flat?), replacing the brick with a long heavy dowel cut
to find one end of the window, you can generate straight-sided ripples.
Add another rod and you can try to get these "traveling ridges" to
interact like solitons (there are higher dimensional versions of the KdV
which model these ridge-like solitons). Try to create a Y shaped
interaction of two of these!
> Philosophically I find them attractive too. In general we seem to find
> physics starts off linear (eg newton gravity and mechanics)
Yrmm... well, when I said "nonlinear" I really meant something more like
this:
1. solitary wavecrest with nonlinear amplitude-dependent velocity,
2. dispersion,
3. dissipation.
Now contrast these two notions:
Phenomenon Examples
(1,2) solitons KdV & Miura equations
(1,3) shocks Burger's equation
> but needs to be refined into increasingly non-linear models (SR, GR) to
> model more extreme situations.
I don't know what you mean by saying str is "nonlinear".
If we accept that "the principle of relativity" states that "fundamental
physical laws are Lorentz invariant", then since the Lorentz group is a
group of affine transformations. Affine transformations are obtained by
combining translations, which can move the origin, with linear
transformations, which cannot, so they are "linear" in the sense that they
take lines to lines, planes to planes.
Or again, in historical context, str gives the correct geometric setting
for Maxwell's theory of EM, which we call "flat spacetime". Gtr
generalizes the good things which follow to curved spacetimes, but
incorporates the "principle of equivalence" in a very simple and
attractive way, which leads to a theory of "gravity as geometry". Very
roughly speaking.
> In particular we see less of "A affects B, but B doesn't affect A" and
> more and more "A affects B, and B affects A". These tend to be in a
> non-linear manner that leads to a certain mathematical complexity.
Maybe you are thinking of Feynman's intuitive explanation (in his
Lectures) of why Maxwell's equations allow the propagation of EM waves?
If so, this is str, and Maxwell's equations are linear in the sense that
taking any linear combination f = c1 f1 + c2 f2 of two old solutions f1,
f2 gives a new solution.
> So the idea that a particle alters the space its in, and space alters
> the particle that's in it is highly attractive as a model.
I think you might be confusing "a theory which admits feedback" with "a
nonlinear theory". But feedback of a kind can be linear; see Feynman's
discussion just cited.
> Solitons offer one neat explanation to particles in that only a limited
> number of possible stable modes might be expected.
Well, I and Gerard mentioned some obvious problems. Another problem is
that the velocity dependence of solitary wave crests in solutions to the
KdV is much more restrictive than particles in mechanics. The real
motivation is more abstract and has to do with the IST as sketched above.
> I also note that EM still remains linear.
Yes.
> At extremes I would expect it to show signs of non-linearity too.
In a sense, in gtr it is linear locally, but in the large, yes, the EM
field energy gravitates and like in gtr all gravitation is nonlinear.
But for weak gravity the nonlinear effects are too small to see, which is
why we got away with Newtonian theory for so long. And human generated EM
fields, or even astrophysical EM fields, probably cannot generate
sufficient densities of field energy to significantly alter ambient
gravitational fields. Too bad, since theoretical solutions like the
Melvin electrovacuum (a magnetic field flux tube held together by its own
self-gravitation) are such fun!
However, on the ArXiv you can find some papers speculating (in more or
less precise mathematical terms, of course!) about "nonlinear EM". Of
course any such theory should reduce to Maxwell for weak fields. (The
theories I am thinking of are not neccessarily intended as "unifications"
of EM with anything, BTW, just as speculations about the possibility that
if we could generate -really- intense EM fields, we'd find something less
familiar to theorists than Einstein-Maxwell.)
> >One obvious problem with trying to relate what I wrote (mostly about
> >the KdV equation) to elementary particle physics is that it is not
> >"relativistic", at least not in any sense that I recognize.
>
> That's a big problem. On the other hand to first order particles are at
> rest with respect to themselves so it might not be quite the problem it
> appears. After all if you (say) obtained a solution for an electronlike
> soliton with three solutions that bore any sort of resemblance to what
> we see, I think that would be a triumph.
Uhm...
> I can't see, though, how to make up a plausible non-linearity. You would
> perhaps be looking for something like gravity that was strong at short
> distances and weak at long distances. That is, behaved like a photon
> (EM) at long distances (but weak) and the weak at short distances (but
> strong).
I understand that both Feynman and Seinfeld have spoken at Cornell, but I
think this is from the latter's presentation :-/
> >Yes, in fact, while the informal term "soliton" apparently does not have a
> >formal definition, everyone agrees that some things count as solitons.
>
> Hmm, suggests that study of solitons has been fraught with difficulties.
It suggests that "soliton theory" is a still a work in progress, but the
"theory" even in its present form has been very successful.
> >Contrast the symmetry Ansatz method I sketched, which enables us to
> >obtain a finite dimensional space of solutions from just one particular
> >solution. While some trivial particular solutions of Burger's equation
> >are evident by inspection, this method is limited by the fact that the
> >point symmetry group of Burger's equation is finite dimensional---
> >clearly, we can obtain only a very small (and highly nonrepresentative)
> >selection of solutions of Burger's equation by this method. OTH, this
> >method applies even when Baecklund morphisms cannot be found.
>
> Of course I have not the faintest idea of the structures you are
> describing, but no matter.
See again what I said about how the IST was discovered.
> My question is whether the mapping is of some solutions of the
> non-linear pde to the linear one.
Other way around: from a solution of the (linear) heat equation, we obtain
a solution of the (nonlinear) Burger's equation.
See above for another point: among other things, I was actually trying to
contrast solitons (conservative, dispersive) with shocks (nonconserative,
dissipative). Two quite different nonlinear phenomena, with two quite
different canonical examples (KdV, Burger's respectively).
> That is we can get some of the non-linear solutions but not all of them.
> Or is it vice versa? Hmm I know you are going to say 'it depends'.....
I was trying to suggest that we might not obtain -all- solutions of
Burger's equation in this way. I.e., our Baecklund morphism (the once
associated with the Hopf-Cole transformation) from the solution space of
the heat equation into the solution space of Burgers' equatation might not
be onto.
> >Yet another variation on this theme which I mentioned is that we can look
> >for the "Lie-Baecklund" symmetry group, which is a supergroup of the point
> >symmetry group, but which may be infinite dimensional even if the point
> >symmetry group is finite dimensional. OTH, it is much harder to compute,
> >and when we obtain the Lie-Baecklund symmetries, it is harder to use them
> >to obtain solutions.
> >
> >Math and mathematical physics are very hard, even in the simplest
> >situations, so tradeoffs abound.
>
> Yes. And the simplicity of linearity must be temptingly attractive.
Well, simplicity -is- very attractive, but what really tempts here is the
fact that the general solution of the heat equation is known from
Sturm-Liouville and Fourier, so there we obtain a huge class of solutions
of Burger's equation (possibly not all solutions of Burger's equation).
> typically, given a mathematical structure modelling physics, (selected)
> solutions to the equations of the model are typically what we observe.
> That is, for example, we often set up a pde to describe movement of a
> body (gravitationally for example), apply boundary conditions
see Sturm-Liouville above.
> and the solution gives us the observed path of the body. I will accept
> that I am likely being naive here.
Maybe you are thinking of Rene Thom's notion of "structural stability"?
> >noone is claiming that you can get away with ignoring nonlinearity.
> >Situations where you can obtain solutions of a particular nonlinear PDE
> >from the known "general solution" of some linear PDE is rather special;
> >see
>
> Indeed. But in a situation where you don't even know the form of the
> non-linear equation and are grasping blindly for a mathematical
> description of experimental results, linear (that is more manageable and
> useful) descriptions are likely to be the first port of call.
Possibly, yes, although one huge lesson of modern dynamical systems theory
is that we now know lots of ways to handle nonlinear models too, and much
better insight into when linear models probably have no hope of cutting
the mustard.
> Then these get modified to tweak a linear system trying to match a
> non-linear one. Then someone hits on the non-linear one....
>
> Which, in general, you can't solve .....
Maybe you are thinking of the fact that a possibly nonlinear gravitation
theory (e.g gtr) must have a Newtonian limit (linear)? If so, the most
interesting examples in say the book by Jackson are arguably completely
new phenomena which did -not- arise from trying to extend some previously
known linear theory. For example, solitons are inherently nonlinear; I
am pretty sure that there is nothing like a weak-field limit of a soliton
which can be treated by a linear theory.
"T. Essel" (annihilistically contemplating crop circle creation)
Note: attempts to locate this bot too precisely will result in large
energy releases in someone's field :-/
tessel@tum.bot
Jul4-04, 07:39 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Mon, 28 Jun 2004, Gerard Westendorp wrote:\n\n> tessel@tum.bot wrote:\n>\n> > The KdV, OTH, admits infinitely many This turns out to be closely\n> > related to the existence of an infinite "integrals of motion"\n> > (corresponding to the conserved quantities mentioned above). This is\n> > characteristic of a "completely integrable Hamiltonian system". (See\n> > again my remarks above concerning n-soliton solutions of the KdV and\n> > conserved quantities.)\n>\n> How would this work out for the linear wave equation?\n\nGood question! Indeed, this very question is one of many items on my\n\n\nAs I trust I did say at various times in my recent posts, I am myself\nstill in the early stages of learning about solitons (among many, many\nother things--- which explains why I am learning this particular material\nso slowly; I know what I have to do, I just can\'t find time enough to do\nit!). As you can probably tell, I have very little background in\ndifferential equations, so in my reading I have come up with many\nquestions which are certainly "natural" and even "fundamental", including\nthis one. The books I am looking at (the ones I cited) are, I think,\nexcellent, but as far as I can see, they do not discuss even briefly the\nquestions on my ever-growing lists. (To be fair, of course I recognize\nthat no author can anticipate every question from the audience, and even\nthat this might not be completely desirable even if it were possible!)\n\nYou raised another question to which I have yet to either figure out the\nanswer on my own, or else to find answered in some book.\n\nNamely: what is "completely integrable" all about, and why oh why are\nthere -two- distinct concepts known by the same name in one field?!\n\nSo far, I have seen two definitions which certainly don\'t look equivalent,\nat least not to me, at least not yet, at least not without thought. The\nwriter who implies that there two different notions are in common use\ndoesn\'t define either one (!), which is why I am not even sure if these\ntwo are the ones this author had in mind. (Mabye the two I know are\nsomehow "obviously equivalent"? -I- don\'t know!)\n\nThe first definition goes by the long name "completely integrable in the\nsense of Liouville", and applies to a n-dimensional Hamiltonian system.\nIf we can find 2n independent conserved integrals which are pairwise\ninvolutory (vanishing Poisson bracket), this system is completely\nintegrable. (I think need some more conditions on these integrals, like\nanalyticity.) This notion is at least recognizably a special case of\n"integrable system".\n\nThe second usage (I hesitate to say "definition") seems silly. Let me\ngive it a long name too: a system is "completely integrable in the sense\nof Lax" if it is solvable by the IST. That is the notion I intended when\nI mentioned "completely integrable" in the place in my post you refer to.\nAccording to my current understanding (I previously tried to clarify this\npoint in my most recent reply to Oz, using different words), "soliton\ntheory" is an undefined term which most researchers take to mean something\nlike "the study of systems which are solvable by the IST or\ngeneralizations of the IST". According to my current understanding, this\nis closely related to "the study of PDEs which possess an infinite\nhierarchy of Lie-Baecklund symmetries".\n\nI meant to say all this in my reply to the OP, but I forgot (IIRC, I was\nwriting under time pressure). Come to think of it, an even better plan\nwould have been to replace "completely integrable" with "is solvable with\nby IST"!.\n\n> My guess: The infinite conserved quantities are just the energies of the\n> linearly independent eigenmodes. These can be rewritten as things like\n> total energy, total momentum, total charge, etc.\n\nI think I can guess why you say that (c.f. the role of eigenmodes in the\nwork of Fermi, Pasta, and Ulam on a failure of equipartition, which I\nthink I mentioned as part of the historical development of soliton\ntheory?), but as yet I don\'t understand this stuff well enough to say if\nyou are right! Actually, this and a similar question have been on one of\nmy lists for some time. The second question is: can\'t we define an\ninfinite sequence of -moments- of the distribution of energy (momentum) in\na solution of say the ordinary 3-dimensional wave equation? My current\nguess: -neither- of these are the same thing at all as an "infinite\nhierarchy" in the sense of KdV!\n\nAs I said, the books I am reading don\'t even mention any of these issues\n(at least not anywhere I have looked so far), so my plan has been to just\ncompute the sequence of things I think I know or can easily figure out how\nto compute for a solution of the wave equation (and a few others), and\nthen to contemplate the results. But I haven\'t gotten around to this yet.\n\nMaybe I should have waited until I understood this stuff better before\nposting, but I went ahead because by the time I get around to improving my\nown (presently rather pitiful) level of understanding, the OP will have\ngone elsewhere, and I wanted to ensure he got a timely response, however\ninadequate.\n\n> Maybe "completely integrable Hamiltonian systems" are a bit like\n> "warped" linear systems, whereas the other ones are not.\n\nDunno--- what are warped linear systems?\n\n> Actually, I do not quite understand what it means physically to have a\n> non-completely integrable system.\n\nIn one of my lists, I have written:\n\nWhat does it mean for a Hamiltonian system to fail to be completely\nintegrable in the sense of Liouville? In the sense of Lax?\n\n(I have so many lists of unanswered questions that I even have lists of\nlists---its awful.)\n\n> Chaos?\n\nI would guess this is not even close (depending, I guess, upon what one\nintends by the famously undefined term "chaos"), but right now I really\nhave no idea!\n\nI hope I can come back eventually and clarify all these points, but for\nnow I can only acknowledge your post and thank you for asking such good\nquestions!\n\n"T. Essel" (hiding somewhere in cyberspace)\n\nP.S.: I have not forgotten that you have a long-time interest in discrete\n<--> continuous, so maybe you are currently in a better position than I to\nexposit for the OP how the Toda lattice fits into the grand scheme of all\nthings KdV? And I myself hope to eventually be in a position to say\nsomething interesting about what Lie would have to say about cellular\nautomata, etc.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Mon, 28 Jun 2004, Gerard Westendorp wrote:
> tessel@tum.bot wrote:
>
> > The KdV, OTH, admits infinitely many This turns out to be closely
> > related to the existence of an infinite "integrals of motion"
> > (corresponding to the conserved quantities mentioned above). This is
> > characteristic of a "completely integrable Hamiltonian system". (See
> > again my remarks above concerning n-soliton solutions of the KdV and
> > conserved quantities.)
>
> How would this work out for the linear wave equation?
Good question! Indeed, this very question is one of many items on my
As I trust I did say at various times in my recent posts, I am myself
still in the early stages of learning about solitons (among many, many
other things--- which explains why I am learning this particular material
so slowly; I know what I have to do, I just can't find time enough to do
it!). As you can probably tell, I have very little background in
differential equations, so in my reading I have come up with many
questions which are certainly "natural" and even "fundamental", including
this one. The books I am looking at (the ones I cited) are, I think,
excellent, but as far as I can see, they do not discuss even briefly the
questions on my ever-growing lists. (To be fair, of course I recognize
that no author can anticipate every question from the audience, and even
that this might not be completely desirable even if it were possible!)
You raised another question to which I have yet to either figure out the
answer on my own, or else to find answered in some book.
Namely: what is "completely integrable" all about, and why oh why are
there -two- distinct concepts known by the same name in one field?!
So far, I have seen two definitions which certainly don't look equivalent,
at least not to me, at least not yet, at least not without thought. The
writer who implies that there two different notions are in common use
doesn't define either one (!), which is why I am not even sure if these
two are the ones this author had in mind. (Mabye the two I know are
somehow "obviously equivalent"? -I- don't know!)
The first definition goes by the long name "completely integrable in the
sense of Liouville", and applies to a n-dimensional Hamiltonian system.
If we can find 2n independent conserved integrals which are pairwise
involutory (vanishing Poisson bracket), this system is completely
integrable. (I think need some more conditions on these integrals, like
analyticity.) This notion is at least recognizably a special case of
"integrable system".
The second usage (I hesitate to say "definition") seems silly. Let me
give it a long name too: a system is "completely integrable in the sense
of Lax" if it is solvable by the IST. That is the notion I intended when
I mentioned "completely integrable" in the place in my post you refer to.
According to my current understanding (I previously tried to clarify this
point in my most recent reply to Oz, using different words), "soliton
theory" is an undefined term which most researchers take to mean something
like "the study of systems which are solvable by the IST or
generalizations of the IST". According to my current understanding, this
is closely related to "the study of PDEs which possess an infinite
hierarchy of Lie-Baecklund symmetries".
I meant to say all this in my reply to the OP, but I forgot (IIRC, I was
writing under time pressure). Come to think of it, an even better plan
would have been to replace "completely integrable" with "is solvable with
by IST"!.
> My guess: The infinite conserved quantities are just the energies of the
> linearly independent eigenmodes. These can be rewritten as things like
> total energy, total momentum, total charge, etc.
I think I can guess why you say that (c.f. the role of eigenmodes in the
work of Fermi, Pasta, and Ulam on a failure of equipartition, which I
think I mentioned as part of the historical development of soliton
theory?), but as yet I don't understand this stuff well enough to say if
you are right! Actually, this and a similar question have been on one of
my lists for some time. The second question is: can't we define an
infinite sequence of -moments- of the distribution of energy (momentum) in
a solution of say the ordinary 3-dimensional wave equation? My current
guess: -neither- of these are the same thing at all as an "infinite
hierarchy" in the sense of KdV!
As I said, the books I am reading don't even mention any of these issues
(at least not anywhere I have looked so far), so my plan has been to just
compute the sequence of things I think I know or can easily figure out how
to compute for a solution of the wave equation (and a few others), and
then to contemplate the results. But I haven't gotten around to this yet.
Maybe I should have waited until I understood this stuff better before
posting, but I went ahead because by the time I get around to improving my
own (presently rather pitiful) level of understanding, the OP will have
gone elsewhere, and I wanted to ensure he got a timely response, however
inadequate.
> Maybe "completely integrable Hamiltonian systems" are a bit like
> "warped" linear systems, whereas the other ones are not.
Dunno--- what are warped linear systems?
> Actually, I do not quite understand what it means physically to have a
> non-completely integrable system.
In one of my lists, I have written:
What does it mean for a Hamiltonian system to fail to be completely
integrable in the sense of Liouville? In the sense of Lax?
(I have so many lists of unanswered questions that I even have lists of
lists---its awful.)
> Chaos?
I would guess this is not even close (depending, I guess, upon what one
intends by the famously undefined term "chaos"), but right now I really
have no idea!
I hope I can come back eventually and clarify all these points, but for
now I can only acknowledge your post and thank you for asking such good
questions!
"T. Essel" (hiding somewhere in cyberspace)
P.S.: I have not forgotten that you have a long-time interest in discrete
<--> continuous, so maybe you are currently in a better position than I to
exposit for the OP how the Toda lattice fits into the grand scheme of all
things KdV? And I myself hope to eventually be in a position to say
something interesting about what Lie would have to say about cellular
automata, etc.
tessel@tum.bot
Jul4-04, 07:39 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Mon, 28 Jun 2004, Gerard Westendorp wrote:\n\n> tessel@tum.bot wrote:\n>\n> > The KdV, OTH, admits infinitely many This turns out to be closely\n> > related to the existence of an infinite "integrals of motion"\n> > (corresponding to the conserved quantities mentioned above). This is\n> > characteristic of a "completely integrable Hamiltonian system". (See\n> > again my remarks above concerning n-soliton solutions of the KdV and\n> > conserved quantities.)\n>\n> How would this work out for the linear wave equation?\n\nGood question! Indeed, this very question is one of many items on my\n\n\nAs I trust I did say at various times in my recent posts, I am myself\nstill in the early stages of learning about solitons (among many, many\nother things--- which explains why I am learning this particular material\nso slowly; I know what I have to do, I just can\'t find time enough to do\nit!). As you can probably tell, I have very little background in\ndifferential equations, so in my reading I have come up with many\nquestions which are certainly "natural" and even "fundamental", including\nthis one. The books I am looking at (the ones I cited) are, I think,\nexcellent, but as far as I can see, they do not discuss even briefly the\nquestions on my ever-growing lists. (To be fair, of course I recognize\nthat no author can anticipate every question from the audience, and even\nthat this might not be completely desirable even if it were possible!)\n\nYou raised another question to which I have yet to either figure out the\nanswer on my own, or else to find answered in some book.\n\nNamely: what is "completely integrable" all about, and why oh why are\nthere -two- distinct concepts known by the same name in one field?!\n\nSo far, I have seen two definitions which certainly don\'t look equivalent,\nat least not to me, at least not yet, at least not without thought. The\nwriter who implies that there two different notions are in common use\ndoesn\'t define either one (!), which is why I am not even sure if these\ntwo are the ones this author had in mind. (Mabye the two I know are\nsomehow "obviously equivalent"? -I- don\'t know!)\n\nThe first definition goes by the long name "completely integrable in the\nsense of Liouville", and applies to a n-dimensional Hamiltonian system.\nIf we can find 2n independent conserved integrals which are pairwise\ninvolutory (vanishing Poisson bracket), this system is completely\nintegrable. (I think need some more conditions on these integrals, like\nanalyticity.) This notion is at least recognizably a special case of\n"integrable system".\n\nThe second usage (I hesitate to say "definition") seems silly. Let me\ngive it a long name too: a system is "completely integrable in the sense\nof Lax" if it is solvable by the IST. That is the notion I intended when\nI mentioned "completely integrable" in the place in my post you refer to.\nAccording to my current understanding (I previously tried to clarify this\npoint in my most recent reply to Oz, using different words), "soliton\ntheory" is an undefined term which most researchers take to mean something\nlike "the study of systems which are solvable by the IST or\ngeneralizations of the IST". According to my current understanding, this\nis closely related to "the study of PDEs which possess an infinite\nhierarchy of Lie-Baecklund symmetries".\n\nI meant to say all this in my reply to the OP, but I forgot (IIRC, I was\nwriting under time pressure). Come to think of it, an even better plan\nwould have been to replace "completely integrable" with "is solvable with\nby IST"!.\n\n> My guess: The infinite conserved quantities are just the energies of the\n> linearly independent eigenmodes. These can be rewritten as things like\n> total energy, total momentum, total charge, etc.\n\nI think I can guess why you say that (c.f. the role of eigenmodes in the\nwork of Fermi, Pasta, and Ulam on a failure of equipartition, which I\nthink I mentioned as part of the historical development of soliton\ntheory?), but as yet I don\'t understand this stuff well enough to say if\nyou are right! Actually, this and a similar question have been on one of\nmy lists for some time. The second question is: can\'t we define an\ninfinite sequence of -moments- of the distribution of energy (momentum) in\na solution of say the ordinary 3-dimensional wave equation? My current\nguess: -neither- of these are the same thing at all as an "infinite\nhierarchy" in the sense of KdV!\n\nAs I said, the books I am reading don\'t even mention any of these issues\n(at least not anywhere I have looked so far), so my plan has been to just\ncompute the sequence of things I think I know or can easily figure out how\nto compute for a solution of the wave equation (and a few others), and\nthen to contemplate the results. But I haven\'t gotten around to this yet.\n\nMaybe I should have waited until I understood this stuff better before\nposting, but I went ahead because by the time I get around to improving my\nown (presently rather pitiful) level of understanding, the OP will have\ngone elsewhere, and I wanted to ensure he got a timely response, however\ninadequate.\n\n> Maybe "completely integrable Hamiltonian systems" are a bit like\n> "warped" linear systems, whereas the other ones are not.\n\nDunno--- what are warped linear systems?\n\n> Actually, I do not quite understand what it means physically to have a\n> non-completely integrable system.\n\nIn one of my lists, I have written:\n\nWhat does it mean for a Hamiltonian system to fail to be completely\nintegrable in the sense of Liouville? In the sense of Lax?\n\n(I have so many lists of unanswered questions that I even have lists of\nlists---its awful.)\n\n> Chaos?\n\nI would guess this is not even close (depending, I guess, upon what one\nintends by the famously undefined term "chaos"), but right now I really\nhave no idea!\n\nI hope I can come back eventually and clarify all these points, but for\nnow I can only acknowledge your post and thank you for asking such good\nquestions!\n\n"T. Essel" (hiding somewhere in cyberspace)\n\nP.S.: I have not forgotten that you have a long-time interest in discrete\n<--> continuous, so maybe you are currently in a better position than I to\nexposit for the OP how the Toda lattice fits into the grand scheme of all\nthings KdV? And I myself hope to eventually be in a position to say\nsomething interesting about what Lie would have to say about cellular\nautomata, etc.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Mon, 28 Jun 2004, Gerard Westendorp wrote:
> tessel@tum.bot wrote:
>
> > The KdV, OTH, admits infinitely many This turns out to be closely
> > related to the existence of an infinite "integrals of motion"
> > (corresponding to the conserved quantities mentioned above). This is
> > characteristic of a "completely integrable Hamiltonian system". (See
> > again my remarks above concerning n-soliton solutions of the KdV and
> > conserved quantities.)
>
> How would this work out for the linear wave equation?
Good question! Indeed, this very question is one of many items on my
As I trust I did say at various times in my recent posts, I am myself
still in the early stages of learning about solitons (among many, many
other things--- which explains why I am learning this particular material
so slowly; I know what I have to do, I just can't find time enough to do
it!). As you can probably tell, I have very little background in
differential equations, so in my reading I have come up with many
questions which are certainly "natural" and even "fundamental", including
this one. The books I am looking at (the ones I cited) are, I think,
excellent, but as far as I can see, they do not discuss even briefly the
questions on my ever-growing lists. (To be fair, of course I recognize
that no author can anticipate every question from the audience, and even
that this might not be completely desirable even if it were possible!)
You raised another question to which I have yet to either figure out the
answer on my own, or else to find answered in some book.
Namely: what is "completely integrable" all about, and why oh why are
there -two- distinct concepts known by the same name in one field?!
So far, I have seen two definitions which certainly don't look equivalent,
at least not to me, at least not yet, at least not without thought. The
writer who implies that there two different notions are in common use
doesn't define either one (!), which is why I am not even sure if these
two are the ones this author had in mind. (Mabye the two I know are
somehow "obviously equivalent"? -I- don't know!)
The first definition goes by the long name "completely integrable in the
sense of Liouville", and applies to a n-dimensional Hamiltonian system.
If we can find 2n independent conserved integrals which are pairwise
involutory (vanishing Poisson bracket), this system is completely
integrable. (I think need some more conditions on these integrals, like
analyticity.) This notion is at least recognizably a special case of
"integrable system".
The second usage (I hesitate to say "definition") seems silly. Let me
give it a long name too: a system is "completely integrable in the sense
of Lax" if it is solvable by the IST. That is the notion I intended when
I mentioned "completely integrable" in the place in my post you refer to.
According to my current understanding (I previously tried to clarify this
point in my most recent reply to Oz, using different words), "soliton
theory" is an undefined term which most researchers take to mean something
like "the study of systems which are solvable by the IST or
generalizations of the IST". According to my current understanding, this
is closely related to "the study of PDEs which possess an infinite
hierarchy of Lie-Baecklund symmetries".
I meant to say all this in my reply to the OP, but I forgot (IIRC, I was
writing under time pressure). Come to think of it, an even better plan
would have been to replace "completely integrable" with "is solvable with
by IST"!.
> My guess: The infinite conserved quantities are just the energies of the
> linearly independent eigenmodes. These can be rewritten as things like
> total energy, total momentum, total charge, etc.
I think I can guess why you say that (c.f. the role of eigenmodes in the
work of Fermi, Pasta, and Ulam on a failure of equipartition, which I
think I mentioned as part of the historical development of soliton
theory?), but as yet I don't understand this stuff well enough to say if
you are right! Actually, this and a similar question have been on one of
my lists for some time. The second question is: can't we define an
infinite sequence of -moments- of the distribution of energy (momentum) in
a solution of say the ordinary 3-dimensional wave equation? My current
guess: -neither- of these are the same thing at all as an "infinite
hierarchy" in the sense of KdV!
As I said, the books I am reading don't even mention any of these issues
(at least not anywhere I have looked so far), so my plan has been to just
compute the sequence of things I think I know or can easily figure out how
to compute for a solution of the wave equation (and a few others), and
then to contemplate the results. But I haven't gotten around to this yet.
Maybe I should have waited until I understood this stuff better before
posting, but I went ahead because by the time I get around to improving my
own (presently rather pitiful) level of understanding, the OP will have
gone elsewhere, and I wanted to ensure he got a timely response, however
inadequate.
> Maybe "completely integrable Hamiltonian systems" are a bit like
> "warped" linear systems, whereas the other ones are not.
Dunno--- what are warped linear systems?
> Actually, I do not quite understand what it means physically to have a
> non-completely integrable system.
In one of my lists, I have written:
What does it mean for a Hamiltonian system to fail to be completely
integrable in the sense of Liouville? In the sense of Lax?
(I have so many lists of unanswered questions that I even have lists of
lists---its awful.)
> Chaos?
I would guess this is not even close (depending, I guess, upon what one
intends by the famously undefined term "chaos"), but right now I really
have no idea!
I hope I can come back eventually and clarify all these points, but for
now I can only acknowledge your post and thank you for asking such good
questions!
"T. Essel" (hiding somewhere in cyberspace)
P.S.: I have not forgotten that you have a long-time interest in discrete
<--> continuous, so maybe you are currently in a better position than I to
exposit for the OP how the Toda lattice fits into the grand scheme of all
things KdV? And I myself hope to eventually be in a position to say
something interesting about what Lie would have to say about cellular
automata, etc.
Gerard Westendorp
Jul7-04, 08:09 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>there have been some interesting tv documentaries lately:\n\n\nhttp://www.bbc.co.uk/science/horizon/2002/freakwave.shtml\n\nApparently, ocean waves of >30meter are much more common\nthan thought possible. They have now been spotted using\nsatellites.\nThese waves are supposedly described by the non-linear\nShrodinger equation...\n\nGerard\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>there have been some interesting tv documentaries lately:
http://www.bbc.co.uk/science/horizon/2002/freakwave.shtml
Apparently, ocean waves of >30meter are much more common
than thought possible. They have now been spotted using
satellites.
These waves are supposedly described by the non-linear
Shrodinger equation...
Gerard
tessel@tum.bot
Jul9-04, 03:49 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Wed, 7 Jul 2004, Gerard Westendorp wrote:\n\n> there have been some interesting tv documentaries lately:\n>\n> http://www.bbc.co.uk/science/horizon/2002/freakwave.shtml\n>\n> Apparently, ocean waves of >30meter are much more common than thought\n> possible. They have now been spotted using satellites.\n\nGosh. But these are -deep- water waves, so I don\'t see how they would\narise from the KdV right now. Did your source say anything about what\nfamous soliton equation is involved?\n\nBe aware that some applied types have a much looser definition than more\ncareful folk. E.g AFAIK "solitons" in the Langmuir lattice are not true\nsolitons. Compare the Toda lattice and the sine-Gordon equation, where we\nhave true solitons.\n\n> These waves are supposedly described by the non-linear Shrodinger\n> equation...\n\nYes, yes, indeed! Gerard, since I last posted here on this topic, I\'ve\nfound some more -fabulous- references which I urgently recommend. Alife\naltering experiences and all that, heh :-/ I\'d like to try to say\nsomething about what\'s in them, but I don\'t have time. Suffice it to say\nthat they answer many of our questions from last time, and raise many new\nand fascinating ones!!!\n\nArghghgh, right now my web browser is broken due to system work in\nprogress, so the complete journal refs and urls are missing below, but I\nhave the ArXiV references:\n\nA beautifully written introduction, from the POV I have been advocating:\n\nauthor = {Richard S. Palais},\ntitle = {The Symmetries of Solitons},\njournal = {Bull. of the A. M. S.}\nvolume = {?},\nyear = {1997}\nnote = {dg-ga/9708004}}\n\nA fabulous book, out of print but available for download at Wood\'s website\n(sorry, due to above problem I can\'t tell you what this is right now),\nwhich complements the book I already cited:\n\neditor = {A. P. Ford and J. C. Wood},\ntitle = {Harmonic Maps and Integrable Systems},\nseries = {Aspects of Mathematics},\nvolume = {E23},\npublisher = {Vieweg},\nyear = 1994,\nnote = {}}\n\nThis book contains excellent articles explaining supremely important\nconnections between solitons and many many things John Baez (and, more\nobsurely alas, myself) have been talking about here in recent years, plus\nharmonic maps (physicists may recognize another buzzword, "nonlinear-sigma\nmodels"), and much more, as well as much more on aspects I\'ve already\ndiscussed such as\n\n* Baecklund morphisms (and the origins of sine-Gordon in classical\ndifferential geometry),\n\n* Lie-Baecklund symmetries, infinite hierarchies of conservation laws and\nhow this relates to the remarkable persistence/stability of solitons,\n\n* Toda lattice models and discrete --> continuous aspects (including\nfurther hints of a connection with special Sturmian tilings, which\napparently Arnold knows all about but which -I- don\'t, arghghg).\n\nThe first article is a fine historical survey which gives an excellent\noverview of IST and how the Schroedinger equation (see above!) enters into\nsoliton theory in a fundamental way.\n\nTo say just one thing about our previous conversation, I should have said\nthat in terms of the range of behavior of nonlinear dynamical systems,\ncompletely integrable systems and "chaotic" systems are in a sense at\nopposite extremes. Completely integrable systems are in a sense maximally\npredictable; chaotic ones are in a sense maximally unpredictable\n(although, unpredictable in highly predictable ways, statistically\nspeaking). Also, from Noether we would expect that the existence of an\ninfinite hierarcy of conservation laws is astonishing indeed, but should\nbe related to extreme stability (all those conserved quantities should\nmake it hard to change things too drastically), and this suspicion turns\nout to be well justified.\n\nRe more on how "chaos" appears in soliton theory:\n\nauthor = {Yanguage (Charles) Li},\ntitle = {Chaos in Partial Differential Equations},\njournal = {Contemporary Mathematics},\nvolume = {?},\nyear = {?},\nnote = {math.AP/0205114}}\n\nFor more on connections with infinite-dimensional Kac-Moody algebras,\ninfinite dimensional Grassmannians, quantum algebra, etc., etc., (see\nabove), and complete integrability:\n\nauthor = {Edward Frenkel},\ntitle = {Five Lectures on Soliton Equations},\nbooktitle = {Surveys in Differential Geometry},\nvolume = 3,\npublisher = {International Press},\nyear = 1997,\nnote = {q-alg./9712005}}\n\nSorry I\'m out of time--- I hope that tomorrow I\'ll be able to try to\nsummarize some of what you can find in these references. I\'d particularly\nlike to say more about\n\n1. Baecklund morphisms and their role in classical differential geometry,\n\n2. a famous mechanical model for the sine-Gordon equation, with links to\nanimated gifs which literally show what I\'d be talking about, and how this\nmay give some intuition for your question regarding how an apparently fine\nbalance between dispersive effects (flattening wavecrests) and nonlinear\ndiffusive effects (steepening wavecrests) can actually be\n"self-adjusting", and thus support -stable- solitary wave solutions.\n\n"T. Essel" (hiding somewhere in cyberspace)\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Wed, 7 Jul 2004, Gerard Westendorp wrote:
> there have been some interesting tv documentaries lately:
>
> http://www.bbc.co.uk/science/horizon/2002/freakwave.shtml
>
> Apparently, ocean waves of >30meter are much more common than thought
> possible. They have now been spotted using satellites.
Gosh. But these are -deep- water waves, so I don't see how they would
arise from the KdV right now. Did your source say anything about what
famous soliton equation is involved?
Be aware that some applied types have a much looser definition than more
careful folk. E.g AFAIK "solitons" in the Langmuir lattice are not true
solitons. Compare the Toda lattice and the sine-Gordon equation, where we
have true solitons.
> These waves are supposedly described by the non-linear Shrodinger
> equation...
Yes, yes, indeed! Gerard, since I last posted here on this topic, I've
found some more -fabulous- references which I urgently recommend. Alife
altering experiences and all that, heh :-/ I'd like to try to say
something about what's in them, but I don't have time. Suffice it to say
that they answer many of our questions from last time, and raise many new
and fascinating ones!!!
Arghghgh, right now my web browser is broken due to system work in
progress, so the complete journal refs and urls are missing below, but I
have the ArXiV references:
A beautifully written introduction, from the POV I have been advocating:
author = {Richard S. Palais},
title = {The Symmetries of Solitons},
journal = {Bull. of the A. M. S.}
volume = {?},
year = {1997}
note = {dg-ga/9708004}}
A fabulous book, out of print but available for download at Wood's website
(sorry, due to above problem I can't tell you what this is right now),
which complements the book I already cited:
editor = {A. P. Ford and J. C. Wood},
title = {Harmonic Maps and Integrable Systems},
series = {Aspects of Mathematics},
volume = {E23},
publisher = {Vieweg},
year = 1994,
note = {}}
This book contains excellent articles explaining supremely important
connections between solitons and many many things John Baez (and, more
obsurely alas, myself) have been talking about here in recent years, plus
harmonic maps (physicists may recognize another buzzword, "nonlinear-\sigma
models"), and much more, as well as much more on aspects I've already
discussed such as
* Baecklund morphisms (and the origins of sine-Gordon in classical
differential geometry),
* Lie-Baecklund symmetries, infinite hierarchies of conservation laws and
how this relates to the remarkable persistence/stability of solitons,
* Toda lattice models and discrete --> continuous aspects (including
further hints of a connection with special Sturmian tilings, which
apparently Arnold knows all about but which -I- don't, arghghg).
The first article is a fine historical survey which gives an excellent
overview of IST and how the Schroedinger equation (see above!) enters into
soliton theory in a fundamental way.
To say just one thing about our previous conversation, I should have said
that in terms of the range of behavior of nonlinear dynamical systems,
completely integrable systems and "chaotic" systems are in a sense at
opposite extremes. Completely integrable systems are in a sense maximally
predictable; chaotic ones are in a sense maximally unpredictable
(although, unpredictable in highly predictable ways, statistically
speaking). Also, from Noether we would expect that the existence of an
infinite hierarcy of conservation laws is astonishing indeed, but should
be related to extreme stability (all those conserved quantities should
make it hard to change things too drastically), and this suspicion turns
out to be well justified.
Re more on how "chaos" appears in soliton theory:
author = {Yanguage (Charles) Li},
title = {Chaos in Partial Differential Equations},
journal = {Contemporary Mathematics},
volume = {?},
year = {?},
note = {math.AP/0205114}}
For more on connections with infinite-dimensional Kac-Moody algebras,
infinite dimensional Grassmannians, quantum algebra, etc., etc., (see
above), and complete integrability:
author = {Edward Frenkel},
title = {Five Lectures on Soliton Equations},
booktitle = {Surveys in Differential Geometry},
volume = 3,
publisher = {International Press},
year = 1997,
note = {q-alg./9712005}}
Sorry I'm out of time--- I hope that tomorrow I'll be able to try to
summarize some of what you can find in these references. I'd particularly
like to say more about
1. Baecklund morphisms and their role in classical differential geometry,
2. a famous mechanical model for the sine-Gordon equation, with links to
animated gifs which literally show what I'd be talking about, and how this
may give some intuition for your question regarding how an apparently fine
balance between dispersive effects (flattening wavecrests) and nonlinear
diffusive effects (steepening wavecrests) can actually be
"self-adjusting", and thus support -stable- solitary wave solutions.
"T. Essel" (hiding somewhere in cyberspace)
tessel@tum.bot
Jul11-04, 02:57 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Hi again, Oz and Gerard, I have some great web references for you!\n\nFirst, Oz, if you are near Snibston Discovery Park, apparently they have\nan outdoor "soliton machine" which recreates the experiments of Scott\nRussell which I described last time! See\n\nhttp://www.ma.hw.ac.uk/solitons/Snibston/index.html\n\nfor some images which illustrate very nicely just what Russell was chasing\non that famous day in 1834.\n\nSecond, Gerard, I have mostly been studying the book edited by Fordy and\nWood which I cited last time, but on a second glance at the paper on\nchaos, I see that Li does mention -deep- water wave solitons which are,\njust as you said, solutions of "the" nonlinear Schrodinger equation\n(obviously, there are many nonlinear variants of the Schrodinger equation;\nI\'d guess this might mean the cubic NLS which comes up in the Toda\nlattice). Naturally, I plan to study this paper closely as soon as I get\na chance.\n\nThird, I think I already mentioned four classic PDEs in this thread:\n\nconvection and dispersion solitons KdV, MKdV, sine-Gordon\n\ndiffusion and dispersion shocks Burger\'s equation\n\nA few days ago I found some web pages with nice animated gifs showing\nsoliton solutions of the first three, comprising, by universal\nacclamation, the three "classic" soliton equations:\n\nhttp://www.math.h.kyoto-u.ac.jp/~takasaki/soliton-lab/gallery/solitons/kdv-e.html\n\nThe page devoted to the sine-Gordon equation shows a mechanical model\nwhich is intended to give some intuition for perhaps the most surprising\nfact about solitons, their amazing persistence/stability. I\'d like to say\nmore about that later.\n\nEnough web references for now! I have learned a great deal in the past\nfew days, so time permitting I\'d like to offer extended elaborations on\nwhat I said before. So, stay tuned for "Solitons in One Thread"...\n\nI\'ll probably run out of time before I can finish even a small part of\nwhat I\'d like to say today, but let me start in and we\'ll see where I wind\nup.\n\nHmm... Baecklund morphisms (or "Baecklund transformations", as they are\ncarelessly called in the literature; don\'t confuse these with\n"Lie-Baeklund symmetries") seem like the simplest topic I could try to\ndiscuss in one post, so let me start with them.\n\nSo, what is a Baecklund transformation? Well, there one example which I\nbet everyone knows--- the Cauchy-Riemann equations! (Sorry, Oz, I guess\nI\'m now addressing my usual grad student and professional audience...).\n\nThe CR equations, as everyone knows, are:\n\nv_x = u_y\n\nv_y = -u_x\n\nThis is a system of first order linear PDEs. But as stated, this system\nis -inconsistent-! That is, if we start with any old u(x,y), say\n\nu = x sin(y^3)\n\nand try to solve for v(x,y), we will derive an inconsistency. To see what\nis going on, differentiate the first equation wrt y and the second wrt x,\nand use the equality of mixed partials:\n\nu_(yy) = v_(xy) = v_(yx) = -u_(xx)\n\nWe can now see that the CR system is consistent iff\n\nu_(xx) + u_(yy) = 0\n\nBut of course this is the whole point of the CR system! The solution\nspace of the two dimensional Laplace equation is called "the space of\nharmonic functions", and the CR equations give a map from that space to\nitself. Specifically, the CR map takes a harmonic function u to the\n(unique, or nearly so) harmonic function v such that\n\nf(x+iy) = u(x,y) + i v(x,y)\n\nis a holomorphic function f:C--> C, i.e. u maps to v chosen such that u,v\nare the real and imaginary parts of a holomorphic function. In complex\nanalysis we say u,v are "conjugate" harmonic functions. As everyone\nknows, conjugate harmonic functions have a beautiful and extremely\nimportant geometric interpretation: the level lines u = u_0 are everywhere\northogonal to the level lines of v = v_0, i.e. we have conformal\ncoordinates u,v on some region of E^2. And they have a beautiful and\nextremely important physical interpretation in terms of two-dimensional\nelectrostatics, and also two-dimensional incompressible fluid flow. For\ndetails, see for example:\n\nauthor = {Tristham Needham},\ntitle = {Visual Complex Analysis},\npublisher = {Oxford University Press},\nyear = 1997}\n\nThe example of the CR system already illustrates the essential character\nof Baecklund automorphisms. We have a space of functions which is the\nspace of solutions to some PDE (often second order and often nonlinear---\nunlike the case here, since Laplace\'s equation is linear), perhaps with\nsome boundary conditions like "rapidly vanishing at spatial infinity".\nThe Baecklund automorphism is a map from this function space to itself,\nand typically is written as a system of PDEs (often first order and\nsometimes linear, as here). But this system is inconsistent without a\n-consistency condition-, which is another PDE (often second order and\noften nonlinear)--- and this PDE is none other than the PDE we started\nwith when we were defining our function space!\n\nHere is another example which will be familiar to gtr fans. Weyl\'s family\nof all static axisymmetric vacuum solutions to the EFE can be written\n\nds^2 = -exp(-2 phi) dt^2\n\n+ exp(2 phi) [ exp(2 psi) (dz^2 + dr^2) + r^2 du^2 ],\n\n-infty < t,z < infty, 0 < r < infty, 0 < u < pi, -pi < v < pi\n\nwhere phi,psi are functions of z,r only and satisfy\n\nphi_(zz) + phi_(rr) + phi_r/r = 0\n\npsi_r = r [(phi_r)^2 - (phi_z)^2]\n\npsi_z = 2 r phi_r phi_z\n\nThis Weyl canonical chart typically only covers an "exterior region".\n\nNote that the definition of the Weyl vacuums includes a system of two\nfirst order PDEs, giving psi(z,r) in terms of phi(z,r), namely:\n\npsi_r = r [(phi_r)^2 - (phi_z)^2]\n\npsi_z = 2 r phi_r phi_z\n\nNever mind the vacuum solutions for the moment and focus attention on this\nsytem. Note that it is inconsistent in general, but by differentiating\nthe first by z and the second by r as above, we derive the consistency\ncondition\n\nphi_(zz) + phi_(rr) + phi_r/r = 0\n\nwhich we recognize as the three-dimensional Laplace equation for an\n-axisymmetric- function, written in cylindrical coordinates. So, here we\nhave a map defined on the space of axisymmetric harmonic functions on R^3\nto another space of functions. By eliminating phi (using differential\nalgebra automates what is otherwise a challenging exercise!), we can find\nthat psi then satisfies another PDE, which happens to be -nonlinear-.\nSo, here we have a Baecklund morphism mapping the space of axisymmetric\nharmonic functions on R^3 into the solution space of a different second\norder PDE.\n\nIt turns out that this second example is getting close to soliton theory!\nAs I said, Weyl\'s family of vacuum solutions to the EFE consists of all\n-static- axisymmetric vacuum solutions. It is generalized by Ernst\'s\nfamily of all -stationary- axisymmetric vacuum solutions. If we consider\nthe problem of finding in closed form some "asympotically flat" Ernst\nvacuums (such as the Kerr vacuum), then methods of soliton theory (IST\nmethod, Hirota method) can be applied to find new solutions from\npreviously known "seed" solutions. Alas, there is a serious defect which\nis apparently shared by all these methods: when fed physically reasonable\n"seeds", these "solution generation methods" seem to often yield less\nreasonable solutions, or at least ones which are hard to interpret\nphysically, at least in the form given by the solution methods. But exact\nsolutions are so hard to come by in gtr that many physicists are happy to\noverlook these problems--- curiously, it is the mathematicians who tend to\nobject!\n\n(Similar remarks hold for the Weyl/Maxwell electrovacuums, the\nBeck/Maxwell electrovacuums, the polarized Gowdy vacuums [colliding plane\nwave models], and their "rotating" or "unpolarized" generalizations, etc.,\nin the possibly unlikely event that any reader knows what those are.)\n\nI don\'t want to deflect attention from the topic at hand (Baecklund\nmorphisms), but I should make one more remark about the Weyl vacuums.\nNote that in the definition of this family of solutions, the consistency\ncondition for the system giving psi(z,r) in terms of phi(z,r) is precisely\nthe condition on phi(z,r) given by the first equation, the\nthree-dimensional axisymmetric Laplace equation. For weak gravitational\nfields, we can show that psi is small, and then the solution reduces to\nthe axisymmetric case of the weak-field approximate solution given by\nEinstein 1915, and the axisymmetric harmonic function phi then agrees\n(possibly after flipping sign) with the axisymmetric Newtonian potential.\nThus, we seemingly have a very simple relationship between a family of\nsolutions in Newtonian gravity and the physically analogous family of\nsolutions in gtr (notice that in both cases we model all static\naxisymmetric gravitational fields, according to Newtonian gravity and\naccording to gtr respectively). Alas, nothing is simple when it comes to\ngtr, and this comment is a case in point. So let me cite without further\ncomment some fine review papers which discuss the Weyl vacuums, and then\nmove on:\n\nauthor = {Jiri Bic\\\'ak},\ntitle = {Selected solutions of {E}instein\'s field equations:\ntheir role in general relativity and astrophysics},\nbooktitle = {{E}instein Field Equations and Their Physical Implications\n(Selected essays in honour of {J}uergen {E}hlers)},\neditor = {Bernd G. Schmidt},\npublisher = {Springer-Verlag},\nyear = 2000,\nseries = {Lecture Notes in Physics},\nvolume = 540,\nnote = {gr-qc/0004016}}\n\nauthor = {W. B. Bonnor},\ntitle = {Physical interpretation of vacuum solutions of {E}instein\'s\nequations. {P}art {I}: {T}ime independent solutions},\njournal = {Gen. Rel. Grav.},\nvolume = 24,\npages = {551--574},\nyear = 1992}\n\nOK, enough about Weyl vacuums. Let\'s move on to a genuine soliton PDE. I\nthink I previously mentioned the sine-Gordon equation\n\nu_(pq) = sin(u)\n\nonly as a "toy" nonlinear wave equation which can be viewed as a variant\nof the Klein-Gordon equation\n\nu_(pq) = u\n\nI think I said that the name "sine-Gordon equation" is a pun, probably\ninvented in the glory days of quantum mechanics by a graduate student who\nlater became famous, George Gamow. (Yes, the author of "the Big Bang",\nanother witty name which has stuck, this one coined, IIRC, not by Gamow\nbut by Fred Hoyle.) Here, I have written both equations in a null\ncoordinate chart for two-dimensional Minkowski spacetime E^(1,1); you can\nrecover the usual Cartesian form by plugging in\n\nx = (q+p)/2 t = (q-p)/2\n\nThe coordinate vector fields have the following geometric character:\n@/@t is timelike, @/@x spacelike, and @/@p, @/@q are null vector fields.\n\nTo put what comes next in context, let me digress again for a moment as\nsay something about wave equations in general. Some of you are probably\nasking, what -is- a wave equation? Well, in the literature, "a two\ndimensional wave equation" seems to usually mean any equation in terms of\nu(x,t) and its derivatives which admits "traveling wave solutions", i.e.\nsolutions of form\n\nu(x,t) = f(kx-ct)\n\nwhere k,c are constants, but the term should probably be regarded as\ninformal, i.e. you can probably get away with using this term any way you\nlike, within reason. Be this as it may, in the case of "wave equations"\nas I defined them--- as I think I mentioned in previous posts--- it is\ncustomary to look for a "dispersion relation", a restriction having the\nform of some algebraic equation (typically) in k,c, which must be\nsatisfied for a traveling wave solution to exist. I\'ll need to say more\nabout dispersion eventually--- but right now I\'m trying to avoid embarking\non subdigressions! So never mind dispersion for the moment.\n\nThe whole point of my expository posts on PDEs has been to popularize\nLie\'s theory of symmetry analysis. Now, one typical game people play when\nthey are doing symmetry analysis is to choose some family of PDEs, say\n\nu_(pq) = K(u)\n\nwhere K is some function of u, and then try to classify special forms of K\nwhich are distinguished by having more or less "point symmetries" (or some\nmore exotic type of symmetry) than the "generic case". One way of looking\nat this procedure is that the results of such a "group analysis" enables\none to quickly pick out from some huge class of PDEs a few specially\nsymmetric cases which are deserving of further study. (In particular, you\nhave a much better chance of finding a large class of solutions to a given\nPDE, via the elementary "symmetry Ansatz method", if your PDE possesses an\nunusually high dimensional point symmetry group).\n\nHmmm... Of course, I should really try to explain what point symmetries\nare before continuing, but that would start a digression within a\ndigression, so for now let me just say that they are transformations of\nthe independent variables (here x,t) and dependent variables (here just u)\nsuch that our PDE is transformed into the same PDE.\n\nThe point of this digression is that I wanted to mention that when we\ncarry out the group analysis for\n\nu_(pq) = K(u)\n\nwe find a handful of distinguished cases, among them:\n\n* K(u) = A exp(Bu) (Liouville equation)\n\n* K(u) = Au + B (Klein-Gordon equation)\n\nThe second class has for its symmetry group a huge infinite dimensional\ngroup G, the first an infinite dimensional subgroup of G, and the generic\ncase is a three dimensional group admitting only the two null translations\n@/@p, @/@q and the books p @/@p - q @/@q. In particular, the sine-Gordon\nequation has only a three dimensional group of point symmetries. Some day\nI\'d like to explain how, in the case of a PDE which like the u_(pq)=K(u)\nadmits a Lagrangian, we can determine the variational symmetry group by\nchecking a simple condition to see which point symmetries are also\nvariational symmetries, and how each of variational symmetry corresponds\nin a "canonical" way to a conserved quantity which typically comes with a\nready-made physical interpretation discovered by inspection, from\nexamining the corresponding flow generating a one-dimensional subgroup of\nthe variational symmetry group. The point here is that this computation\nis in a way terribly misleading in the case of soltion equations; they\ntypically come with a handful of ready-made conserved quantities, but\nactually have an infinite hierarchy of them! These extra conserved\nquantities are "hidden from Lie\'s view" and thus are called "hidden\nsymmetries". Fortunately, there are straightforward techniques which can\noften uncover hidden symmetries, when they exist.\n\nSome day I\'d like to discuss this group analysis in detail, plus that of\nother possible "wave equations", such as a particular favorite of mine:\n\nu_(pq) = K(u) u_p u_q\n\nOK, end of digression. Back to Baecklund transformations. I was talking\nabout the sine-Gordon equation and I\'d like to explain how this equation\narose in classical differential geometry many decades before quantum\nmechanics and the Klein-Gordon equation came along. I find the\nexplanations I\'ve seen make this look much too hard, so I\'ll try for a\nmore direct approach.\n\nBut first a bit of historical background. Noneuclidean geometry in\ngeneral, and hyperbolic geometry in particular, was a very hot topic in\nthe mid nineteenth century. Mathematicians were still trying to convince\npeople that H^2 even -exists- (Lewis Carroll poked fun at this debate, and\na whole lot of further nineteenth century mathematics, in his Alice\nstories). One approach to the problem of making noneuclidean geometry\n"more intuitive" is to search for embedded surfaces in E^3 (not\n"immersed", if you know what that is; read on) whose induced metrics have\na Gaussian curvature with constant value -1. Beltrami found the first\nexample, but there are infinitely many, and these surfaces can be\nextremely beautiful, so there is also an aesthetic motivation. These\nsurfaces also clearly exhibit the profound distinction between -local- and\n-global- isometries; they all have constant Gaussian curvature -1, so they\nare all -locally- isometric to H^2, but not globally so.\n\nSo, one thing a French geometer like Darboux might be doing circa 1870\nwould be trying to solve this problem: find all asymptotic coordinate\ncharts describing the intrinsic geometry of surfaces which have constant\nGaussian curvature -1, i.e. surface which are locally isometric to the\nhyperbolic plane. Here, "asymptotic lines" are a special kind of curves\non a given surface which get a lot of press in classical differential\ngeometry (IIRC, they are discussed with a vivid illustration in Hilbert\nand Cohn-Vossen, Geometry and the Imagination, Chelsea). As you might\nguess, the "coordinate lines" of an "asymptotic coordinate chart" are\n"asymptotic lines"; hence the name. In this case, it turns out that\nsolving the problem just mentioned yields embeddings of surfaces locally\nisometric to H^2 in a natural way. BTW, I could have used a different\nkind of chart here instead, the Chebyshev chart (yes, this is the\nChebyshev of polynomial fame, and also of probability and number theory\nfame), or any of a number of other special types of charts; all were in\nfact considered by nineteenth century geometers.\n\nBut never mind for now exactly how we can produce embedded surfaces from a\nspecial chart such as an asymptotic chart; let me just say what an\nasymptotic coordinate chart -is-. Fortunately, this is very easy! An\nasymptotic coordinate chart is any chart having the form\n\nds^2 = dp^2 + 2 cos(u) dp dq + dq^2\n\nwhere u is some function of p,q. Depending on the choice of u, we\'ll\nobtain different geodesics and different Gaussian curvatures, i.e.\ndifferent intrinsic geometries in the sense of surface theory.\nIronically, this is now much easier to explain than the -extrinsic\ngeometry- which I am sweeping under the rug, even though to a nineteenth\ncentury geometer it would be pretty much the point of this exercise. I\nhowever have a different goal--- I want to show how the sine-Gordon\nequation arises very naturally when you are studying asymptotic coordinate\ncharts.\n\nAnachronistically following Elie Cartan, let\'s introduce an obvious\ncoframe\n\no^1 = dp + cos(u) dq\n\no^2 = sin(u) dq\n\nYou can check that the metric tensor\n\no^1 & o^1 + o^2 & o^2\n\n(tensor product!) gives us back the line element above. Next, computing\n\ndo^1 = -sin(u) u_p dp /\\ dq\n\n= -u_p dp /\\ sin(u) dq\n\n= -w^1_2 /\\ o^2\n\nwe see that the nontrivial connection 1-forms are given (up to algebraic\nsymmetry) by\n\nw^1_2 = u_p dp\n\nNext,\n\ndw^1_2 = u_(pq) dq /\\ dp\n\n= -u_(pq) dp /\\ dq\n\n= -u_(pq)/sin(u) o^1 /\\ o^2\n\nThus the Riemann tensor is given (up to algebraic symmetry) by\n\nR^1_(212) = -u_(pq)/sin(u)\n\nwhich is also the Gaussian curvature scalar. And of course this will have\nconstant value -1 iff u(p,q) satisfies\n\nu_(pq) = sin(u)\n\n---which is the sine-Gordon equation!\n\n(This would be a good moment to look at the web page cited above offering\na few animated gifs of a mechanical model for the sine-Gordon equation.\nYou probably wouldn\'t have guessed from that rod and rubber band\ncontraption that the controlling equation had anything to do with\ndifferential geometry-- but now you know that it -does-!)\n\nNotice that something rather remarkable happened here (not for the first\ntime, nor the last; amusingly--- and delightfully--- in solition theory,\n"remarkable phenomena" turn out to be -ubiquitous-!). We started looking\nfor a condition on a special kind of coordinate chart which ensures that\nour surface is locally isometric to H^2, a Riemannian manifold, but the\ncondition we found was a PDE which, from another point of view, can be\ninterpreted as a nonlinear wave equation on E^(1,1), a zero-curvature\nsemi-Riemannian manifold! What does this mean?\n\nWell, it means we have found a harmonic map. Alas, I don\'t have time to\nexplain what harmonic maps are right now (but Charlie Torre can, if he\'s\nreading this). However, if I were to say just one thing about harmonic\nmaps, I guess it would have to be this: in a sense, they define -surfaces-\nwhich generalize -geodesic curves- in the sense that these surfaces\nminimize a certain "energy integral", in way which is geometrically\nanalogous to minimizing length. Examples of harmonic maps arise in\nembedding "space forms" such as H^2 and in embedding "minimal surfaces"\n(this requires harmonic functions, as most readers will know); readers\nfamiliar with differential geometry will know this involves looking for\nembeddings of surfaces which obey constraints on intrinsic and extrinsic\ncurvature, respectively. But harmonic maps arise in other contexts. For\nexample, the so-called "nonlinear sigma models" in physics are defined in\nterms of harmonic maps; in fact, there is an ambitious program underway to\nexpress many of the most important equations of modern physics in terms of\nharmonic maps! The point is that this topic is closely related to soliton\ntheory, because it turns out that the same methods mentioned above in the\ncontext of solving the Cauchy problem for solition PDEs also can be\nemployed to find harmonic maps! And--- again, quite remarkably--- the\nphrase "zero-curvature" will arise again in an essential way, should I\never get to discussing the IST!\n\nIn a later post, I may get a chance to return to the hierarchy of\nconservation laws of the KdV (and a closely related phenomenon, the\nhierarchy of Lie-Baecklund symmetries). If I ever get to discussing the\nIST, I\'ll certainly need to exhibit the Baecklund morphism mapping between\nthe KdV and Miura\'s equation (AKA the MKdV equation), and discuss how this\nthe conserved quantities of the image and target solutions are related,\nbecause this is needed to even outline how the IST works. For now, I just\nwant to correct an pedagogical oversight.\n\nWhen I previously discussed the "conserved densities" and "fluxes", I\nthink I neglected to point out a terribly important feature: they are all\n-differential polynomials-, i.e. polynomials in u, u_x, u_(xx), etc. For\nexample, for the KdV the first three densities are\n\nu\n\nu^2/2\n\nu^3 - (u_x)^2/2\n\nThe point is, for some time I have been threatening to back way up and\ntalk about Lie analysis of (systems of) ODEs, where we are very quickly\nled to Kleinian geometries acted upon by Lie groups, and differential\npolynomial invariants which typically look much like these!\n\nIf only I had previously found time to show how this works (it\'s not\nneeded, perhaps, for a first look at solitons, but it is an -extremely-\nbeautiful topic in its own right), I could now try to develop a new theme:\nordinary integrable systems (say tractable systems of ODEs) typically\nsecretly involve a finite dimensional Kleinian geometry, which is\nreflected in the existence of a finite number of independent and\nnontrivial conserved quantities (typically interpretable as things like\nenergy and momentum); if you are sufficiently mad, you might now guess\nthat completely integrable systems of PDEs typically secretly involve an\n-infinite dimensional- Kleinian geometry, which is reflected in the\nexistence of an infinite number of independent and nontrivial conserved\nquantities. Utterly remarkably, you would be correct!\n\nBut that\'s just the tip of the iceberg--- these Kleinian geometries\ntypically are in fact some kind of "algebraic variety"! In particular,\nthe infinite hierarchy of conservation laws of the KdV secretly reflect\nstuff happening on an -infinite dimensional Grassmannian-! But seeing how\nthis works involves learning a fair amount about invariant theory, Lie\nalgebras (and Kac-Moody algebras), algebraic geometry, symmetry analysis\nof differential equations, Kleinian and Cartan geometry, all of which are\nhuge topics in their own right, so at best I can hope only to, in the\nfuture, throw out more elaborate hints than I am right at this moment.\n\nI mention this missed opportunity because of a historical phenomenon which\nkeeps impressing me more and more strongly as I learn more about\ncontemporary solition theory and Lie analysis. In previous posts I have\nmentioned the visit to Paris in 1870 of two young mathematicians studying\nin Berlin, Felix Klein and Sophus Lie. Among the people they talked to\nwas Gaston Darboux, the inventor of "Darboux transformations", a kind of\ngeneralization of Baecklund transformations. I have previously tried to\nsketch how the crucible of the early ideas of Klein on "Kleinian geometry"\nand the early ideas of Lie on Lie analysis of differential equations, Lie\nalgebras, and Lie groups, was--- algebraic geometry! It is interesting\nthat soliton theorists seem to currently be uncovering the algebraic roots\nof their subject! If I forget to return to this theme in future posts, I\nhope someone will remind me, because I think it\'s important.\n\nOh dear, once again I\'m out of time and so forth, and I never even got to\nthe Baecklund morphisms for the sine-Gordon equation, much less the KdV!\nOh well, more later, access permitting.\n\n"T. Essel" (hiding somewhere in cyberspace)\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hi again, Oz and Gerard, I have some great web references for you!
First, Oz, if you are near Snibston Discovery Park, apparently they have
an outdoor "soliton machine" which recreates the experiments of Scott
Russell which I described last time! See
http://www.ma.hw.ac.uk/solitons/Snibston/index.html
for some images which illustrate very nicely just what Russell was chasing
on that famous day in 1834.
Second, Gerard, I have mostly been studying the book edited by Fordy and
Wood which I cited last time, but on a second glance at the paper on
chaos, I see that Li does mention -deep- water wave solitons which are,
just as you said, solutions of "the" nonlinear Schrodinger equation
(obviously, there are many nonlinear variants of the Schrodinger equation;
I'd guess this might mean the cubic NLS which comes up in the Toda
lattice). Naturally, I plan to study this paper closely as soon as I get
a chance.
Third, I think I already mentioned four classic PDEs in this thread:
convection and dispersion solitons KdV, MKdV, sine-Gordon
diffusion and dispersion shocks Burger's equation
A few days ago I found some web pages with nice animated gifs showing
soliton solutions of the first three, comprising, by universal
acclamation, the three "classic" soliton equations:
http://www.math.h.kyoto-u.ac.jp/~takasaki/soliton-lab/gallery/solitons/kdv-e.html
The page devoted to the sine-Gordon equation shows a mechanical model
which is intended to give some intuition for perhaps the most surprising
fact about solitons, their amazing persistence/stability. I'd like to say
more about that later.
Enough web references for now! I have learned a great deal in the past
few days, so time permitting I'd like to offer extended elaborations on
what I said before. So, stay tuned for "Solitons in One Thread"...
I'll probably run out of time before I can finish even a small part of
what I'd like to say today, but let me start in and we'll see where I wind
up.
Hmm... Baecklund morphisms (or "Baecklund transformations", as they are
carelessly called in the literature; don't confuse these with
"Lie-Baeklund symmetries") seem like the simplest topic I could try to
discuss in one post, so let me start with them.
So, what is a Baecklund transformation? Well, there one example which I
bet everyone knows--- the Cauchy-Riemann equations! (Sorry, Oz, I guess
I'm now addressing my usual grad student and professional audience...).
The CR equations, as everyone knows, are:
v_x = u_yv_y = -u_x
This is a system of first order linear PDEs. But as stated, this system
is -inconsistent-! That is, if we start with any old u(x,y), say
u = x sin(y^3)
and try to solve for v(x,y), we will derive an inconsistency. To see what
is going on, differentiate the first equation wrt y and the second wrt x,
and use the equality of mixed partials:
u_(yy) = v_(xy) = v_(yx) = -u_(xx)
We can now see that the CR system is consistent iff
u_(xx) + u_(yy) =
But of course this is the whole point of the CR system! The solution
space of the two dimensional Laplace equation is called "the space of
harmonic functions", and the CR equations give a map from that space to
itself. Specifically, the CR map takes a harmonic function u to the
(unique, or nearly so) harmonic function v such that
f(x+iy) = u(x,y) + i v(x,y)
is a holomorphic function f:C--> C, i.e. u maps to v chosen such that u,v
are the real and imaginary parts of a holomorphic function. In complex
analysis we say u,v are "conjugate" harmonic functions. As everyone
knows, conjugate harmonic functions have a beautiful and extremely
important geometric interpretation: the level lines u = u_0 are everywhere
orthogonal to the level lines of v = v_0, i.e. we have conformal
coordinates u,v on some region of E^2. And they have a beautiful and
extremely important physical interpretation in terms of two-dimensional
electrostatics, and also two-dimensional incompressible fluid flow. For
details, see for example:
author = {Tristham Needham},
title = {Visual Complex Analysis},
publisher = {Oxford University Press},
year = 1997}
The example of the CR system already illustrates the essential character
of Baecklund automorphisms. We have a space of functions which is the
space of solutions to some PDE (often second order and often nonlinear---
unlike the case here, since Laplace's equation is linear), perhaps with
some boundary conditions like "rapidly vanishing at spatial infinity".
The Baecklund automorphism is a map from this function space to itself,
and typically is written as a system of PDEs (often first order and
sometimes linear, as here). But this system is inconsistent without a
-consistency condition-, which is another PDE (often second order and
often nonlinear)--- and this PDE is none other than the PDE we started
with when we were defining our function space!
Here is another example which will be familiar to gtr fans. Weyl's family
of all static axisymmetric vacuum solutions to the EFE can be written
ds^2 = -\exp(-2 \phi) dt^2+ \exp(2 \phi) [ \exp(2 \psi) (dz^2 + dr^2) + r^2 du^2 ],-\infty <[/itex] t,z < \infty,< r < \infty,< u < \pi, -\pi < v < \pi
where \phi,\psi are functions of z,r only and satisfy
\phi_(zz) + \phi_(rr) + \phi_r/r = \psi_r = r [(\phi_r)^2 - (\phi_z)^2]\psi_z = 2 r \phi_r \phi_z
This Weyl canonical chart typically only covers an "exterior region".
Note that the definition of the Weyl vacuums includes a system of two
first order PDEs, giving \psi(z,r) in terms of \phi(z,r), namely:
\psi_r = r [(\phi_r)^2 - (\phi_z)^2]\psi_z = 2 r \phi_r \phi_z
Never mind the vacuum solutions for the moment and focus attention on this
sytem. Note that it is inconsistent in general, but by differentiating
the first by z and the second by r as above, we derive the consistency
condition
\phi_(zz) + \phi_(rr) + \phi_r/r =
which we recognize as the three-dimensional Laplace equation for an
-axisymmetric- function, written in cylindrical coordinates. So, here we
have a map defined on the space of axisymmetric harmonic functions on R^3
to another space of functions. By eliminating \phi (using differential
algebra automates what is otherwise a challenging exercise!), we can find
that \psi then satisfies another PDE, which happens to be -nonlinear-.
So, here we have a Baecklund morphism mapping the space of axisymmetric
harmonic functions on R^3 into the solution space of a different second
order PDE.
It turns out that this second example is getting close to soliton theory!
As I said, Weyl's family of vacuum solutions to the EFE consists of all
-static- axisymmetric vacuum solutions. It is generalized by Ernst's
family of all -stationary- axisymmetric vacuum solutions. If we consider
the problem of finding in closed form some "asympotically flat" Ernst
vacuums (such as the Kerr vacuum), then methods of soliton theory (IST
method, Hirota method) can be applied to find new solutions from
previously known "seed" solutions. Alas, there is a serious defect which
is apparently shared by all these methods: when fed physically reasonable
"seeds", these "solution generation methods" seem to often yield less
reasonable solutions, or at least ones which are hard to interpret
physically, at least in the form given by the solution methods. But exact
solutions are so hard to come by in gtr that many physicists are happy to
overlook these problems--- curiously, it is the mathematicians who tend to
object!
(Similar remarks hold for the Weyl/Maxwell electrovacuums, the
Beck/Maxwell electrovacuums, the polarized Gowdy vacuums [colliding plane
wave models], and their "rotating" or "unpolarized" generalizations, etc.,
in the possibly unlikely event that any reader knows what those are.)
I don't want to deflect attention from the topic at hand (Baecklund
morphisms), but I should make one more remark about the Weyl vacuums.
Note that in the definition of this family of solutions, the consistency
condition for the system giving \psi(z,r) in terms of \phi(z,r) is precisely
the condition on \phi(z,r) given by the first equation, the
three-dimensional axisymmetric Laplace equation. For weak gravitational
fields, we can show that \psi is small, and then the solution reduces to
the axisymmetric case of the weak-field approximate solution given by
Einstein 1915, and the axisymmetric harmonic function \phi then agrees
(possibly after flipping sign) with the axisymmetric Newtonian potential.
Thus, we seemingly have a very simple relationship between a family of
solutions in Newtonian gravity and the physically analogous family of
solutions in gtr (notice that in both cases we model all static
axisymmetric gravitational fields, according to Newtonian gravity and
according to gtr respectively). Alas, nothing is simple when it comes to
gtr, and this comment is a case in point. So let me cite without further
comment some fine review papers which discuss the Weyl vacuums, and then
move on:
author = {Jiri Bic\'ak},
title = {Selected solutions of {E}instein's field equations:
their role in general relativity and astrophysics},
booktitle = {{E}instein Field Equations and Their Physical Implications
(Selected essays in honour of {J}uergen {E}hlers)},
editor = {Bernd G. Schmidt},
publisher = {Springer-Verlag},
year = 2000,
series = {Lecture Notes in Physics},
volume = 540,
note = {http://www.arxiv.org/abs/gr-qc/0004016}}
author = {W. B. Bonnor},
title = {Physical interpretation of vacuum solutions of {E}instein's
equations. {P}art {I}: {T}ime independent solutions},
journal = {Gen. Rel. Grav.},
volume = 24,
pages = {551--574},
year = 1992}
OK, enough about Weyl vacuums. Let's move on to a genuine soliton PDE. I
think I previously mentioned the sine-Gordon equation
u_(pq) = sin(u)
only as a "toy" nonlinear wave equation which can be viewed as a variant
of the Klein-Gordon equation
u_(pq) = u
I think I said that the name "sine-Gordon equation" is a pun, probably
invented in the glory days of quantum mechanics by a graduate student who
later became famous, George Gamow. (Yes, the author of "the Big Bang",
another witty name which has stuck, this one coined, IIRC, not by Gamow
but by Fred Hoyle.) Here, I have written both equations in a null
coordinate chart for two-dimensional Minkowski spacetime E^(1,1); you can
recover the usual Cartesian form by plugging in
x = (q+p)/2 t = (q-p)/2
The coordinate vector fields have the following geometric character:
@/@t is timelike, @/@x spacelike, and @/@p, @/@q are null vector fields.
To put what comes next in context, let me digress again for a moment as
say something about wave equations in general. Some of you are probably
asking, what -is- a wave equation? Well, in the literature, "a two
dimensional wave equation" seems to usually mean any equation in terms of
u(x,t) and its derivatives which admits "traveling wave solutions", i.e.
solutions of form
u(x,t) = f(kx-ct)
where k,c are constants, but the term should probably be regarded as
informal, i.e. you can probably get away with using this term any way you
like, within reason. Be this as it may, in the case of "wave equations"
as I defined them--- as I think I mentioned in previous posts--- it is
customary to look for a "dispersion relation", a restriction having the
form of some algebraic equation (typically) in k,c, which must be
satisfied for a traveling wave solution to exist. I'll need to say more
about dispersion eventually--- but right now I'm trying to avoid embarking
on subdigressions! So never mind dispersion for the moment.
The whole point of my expository posts on PDEs has been to popularize
Lie's theory of symmetry analysis. Now, one typical game people play when
they are doing symmetry analysis is to choose some family of PDEs, say
u_(pq) = K(u)
where K is some function of u, and then try to classify special forms of K
which are distinguished by having more or less "point symmetries" (or some
more exotic type of symmetry) than the "generic case". One way of looking
at this procedure is that the results of such a "group analysis" enables
one to quickly pick out from some huge class of PDEs a few specially
symmetric cases which are deserving of further study. (In particular, you
have a much better chance of finding a large class of solutions to a given
PDE, via the elementary "symmetry Ansatz method", if your PDE possesses an
unusually high dimensional point symmetry group).
Hmmm... Of course, I should really try to explain what point symmetries
are before continuing, but that would start a digression within a
digression, so for now let me just say that they are transformations of
the independent variables (here x,t) and dependent variables (here just u)
such that our PDE is transformed into the same PDE.
The point of this digression is that I wanted to mention that when we
carry out the group analysis for
u_(pq) = K(u)
we find a handful of distinguished cases, among them:
* K(u) = A \exp(Bu) (Liouville equation)
* K(u) = Au + B (Klein-Gordon equation)
The second class has for its symmetry group a huge infinite dimensional
group G, the first an infinite dimensional subgroup of G, and the generic
case is a three dimensional group admitting only the two null translations
@/@p, @/@q and the books p @/@p - q @/@q. In particular, the sine-Gordon
equation has only a three dimensional group of point symmetries. Some day
I'd like to explain how, in the case of a PDE which like the u_(pq)=K(u)
admits a Lagrangian, we can determine the variational symmetry group by
checking a simple condition to see which point symmetries are also
variational symmetries, and how each of variational symmetry corresponds
in a "canonical" way to a conserved quantity which typically comes with a
ready-made physical interpretation discovered by inspection, from
examining the corresponding flow generating a one-dimensional subgroup of
the variational symmetry group. The point here is that this computation
is in a way terribly misleading in the case of soltion equations; they
typically come with a handful of ready-made conserved quantities, but
actually have an infinite hierarchy of them! These extra conserved
quantities are "hidden from Lie's view" and thus are called "hidden
symmetries". Fortunately, there are straightforward techniques which can
often uncover hidden symmetries, when they exist.
Some day I'd like to discuss this group analysis in detail, plus that of
other possible "wave equations", such as a particular favorite of mine:
[itex]u_(pq) = K(u) u_p u_q
OK, end of digression. Back to Baecklund transformations. I was talking
about the sine-Gordon equation and I'd like to explain how this equation
arose in classical differential geometry many decades before quantum
mechanics and the Klein-Gordon equation came along. I find the
explanations I've seen make this look much too hard, so I'll try for a
more direct approach.
But first a bit of historical background. Noneuclidean geometry in
general, and hyperbolic geometry in particular, was a very hot topic in
the mid nineteenth century. Mathematicians were still trying to convince
people that H^2 even -exists- (Lewis Carroll poked fun at this debate, and
a whole lot of further nineteenth century mathematics, in his Alice
stories). One approach to the problem of making noneuclidean geometry
"more intuitive" is to search for embedded surfaces in E^3 (not
"immersed", if you know what that is; read on) whose induced metrics have
a Gaussian curvature with constant value -1. Beltrami found the first
example, but there are infinitely many, and these surfaces can be
extremely beautiful, so there is also an aesthetic motivation. These
surfaces also clearly exhibit the profound distinction between -local- and
-global- isometries; they all have constant Gaussian curvature -1, so they
are all -locally- isometric to H^2, but not globally so.
So, one thing a French geometer like Darboux might be doing circa 1870
would be trying to solve this problem: find all asymptotic coordinate
charts describing the intrinsic geometry of surfaces which have constant
Gaussian curvature -1, i.e. surface which are locally isometric to the
hyperbolic plane. Here, "asymptotic lines" are a special kind of curves
on a given surface which get a lot of press in classical differential
geometry (IIRC, they are discussed with a vivid illustration in Hilbert
and Cohn-Vossen, Geometry and the Imagination, Chelsea). As you might
guess, the "coordinate lines" of an "asymptotic coordinate chart" are
"asymptotic lines"; hence the name. In this case, it turns out that
solving the problem just mentioned yields embeddings of surfaces locally
isometric to H^2 in a natural way. BTW, I could have used a different
kind of chart here instead, the Chebyshev chart (yes, this is the
Chebyshev of polynomial fame, and also of probability and number theory
fame), or any of a number of other special types of charts; all were in
fact considered by nineteenth century geometers.
But never mind for now exactly how we can produce embedded surfaces from a
special chart such as an asymptotic chart; let me just say what an
asymptotic coordinate chart -is-. Fortunately, this is very easy! An
asymptotic coordinate chart is any chart having the form
ds^2 = dp^2 + 2 cos(u) dp dq + dq^2
where u is some function of p,q. Depending on the choice of u, we'll
obtain different geodesics and different Gaussian curvatures, i.e.
different intrinsic geometries in the sense of surface theory.
Ironically, this is now much easier to explain than the -extrinsic
geometry- which I am sweeping under the rug, even though to a nineteenth
century geometer it would be pretty much the point of this exercise. I
however have a different goal--- I want to show how the sine-Gordon
equation arises very naturally when you are studying asymptotic coordinate
charts.
Anachronistically following Elie Cartan, let's introduce an obvious
coframe
o^1 = dp + cos(u) dqo^2 = sin(u) dq
You can check that the metric tensor
o^1 & o^1 + o^2 & o^2
(tensor product!) gives us back the line element above. Next, computing
do^1 = -sin(u) u_p dp /\ dq= -u_p dp /\ sin(u) dq= -w^{1_2} /\ o^2
we see that the nontrivial connection 1-forms are given (up to algebraic
symmetry) by
w^{1_2} = u_p dp
Next,
dw^1_2 = u_(pq) dq /\ dp= -u_(pq) dp /\ dq= -u_(pq)/sin(u) o^1 /\ o^2
Thus the Riemann tensor is given (up to algebraic symmetry) by
R^{1_}(212) = -u_(pq)/sin(u)
which is also the Gaussian curvature scalar. And of course this will have
constant value -1 iff u(p,q) satisfies
u_(pq) = sin(u)
---which is the sine-Gordon equation!
(This would be a good moment to look at the web page cited above offering
a few animated gifs of a mechanical model for the sine-Gordon equation.
You probably wouldn't have guessed from that rod and rubber band
contraption that the controlling equation had anything to do with
differential geometry-- but now you know that it -does-!)
Notice that something rather remarkable happened here (not for the first
time, nor the last; amusingly--- and delightfully--- in solition theory,
"remarkable phenomena" turn out to be -ubiquitous-!). We started looking
for a condition on a special kind of coordinate chart which ensures that
our surface is locally isometric to H^2, a Riemannian manifold, but the
condition we found was a PDE which, from another point of view, can be
interpreted as a nonlinear wave equation on E^(1,1), a zero-curvature
semi-Riemannian manifold! What does this mean?
Well, it means we have found a harmonic map. Alas, I don't have time to
explain what harmonic maps are right now (but Charlie Torre can, if he's
reading this). However, if I were to say just one thing about harmonic
maps, I guess it would have to be this: in a sense, they define -surfaces-
which generalize -geodesic curves- in the sense that these surfaces
minimize a certain "energy integral", in way which is geometrically
analogous to minimizing length. Examples of harmonic maps arise in
embedding "space forms" such as H^2 and in embedding "minimal surfaces"
(this requires harmonic functions, as most readers will know); readers
familiar with differential geometry will know this involves looking for
embeddings of surfaces which obey constraints on intrinsic and extrinsic
curvature, respectively. But harmonic maps arise in other contexts. For
example, the so-called "nonlinear \sigma models" in physics are defined in
terms of harmonic maps; in fact, there is an ambitious program underway to
express many of the most important equations of modern physics in terms of
harmonic maps! The point is that this topic is closely related to soliton
theory, because it turns out that the same methods mentioned above in the
context of solving the Cauchy problem for solition PDEs also can be
employed to find harmonic maps! And--- again, quite remarkably--- the
phrase "zero-curvature" will arise again in an essential way, should I
ever get to discussing the IST!
In a later post, I may get a chance to return to the hierarchy of
conservation laws of the KdV (and a closely related phenomenon, the
hierarchy of Lie-Baecklund symmetries). If I ever get to discussing the
IST, I'll certainly need to exhibit the Baecklund morphism mapping between
the KdV and Miura's equation (AKA the MKdV equation), and discuss how this
the conserved quantities of the image and target solutions are related,
because this is needed to even outline how the IST works. For now, I just
want to correct an pedagogical oversight.
When I previously discussed the "conserved densities" and "fluxes", I
think I neglected to point out a terribly important feature: they are all
-differential polynomials-, i.e. polynomials in u, u_x, u_(xx), etc. For
example, for the KdV the first three densities are
u
u^2/2u^3 - (u_x)^2/2
The point is, for some time I have been threatening to back way up and
talk about Lie analysis of (systems of) ODEs, where we are very quickly
led to Kleinian geometries acted upon by Lie groups, and differential
polynomial invariants which typically look much like these!
If only I had previously found time to show how this works (it's not
needed, perhaps, for a first look at solitons, but it is an -extremely-
beautiful topic in its own right), I could now try to develop a new theme:
ordinary integrable systems (say tractable systems of ODEs) typically
secretly involve a finite dimensional Kleinian geometry, which is
reflected in the existence of a finite number of independent and
nontrivial conserved quantities (typically interpretable as things like
energy and momentum); if you are sufficiently mad, you might now guess
that completely integrable systems of PDEs typically secretly involve an
-infinite dimensional- Kleinian geometry, which is reflected in the
existence of an infinite number of independent and nontrivial conserved
quantities. Utterly remarkably, you would be correct!
But that's just the tip of the iceberg--- these Kleinian geometries
typically are in fact some kind of "algebraic variety"! In particular,
the infinite hierarchy of conservation laws of the KdV secretly reflect
stuff happening on an -infinite dimensional Grassmannian-! But seeing how
this works involves learning a fair amount about invariant theory, Lie
algebras (and Kac-Moody algebras), algebraic geometry, symmetry analysis
of differential equations, Kleinian and Cartan geometry, all of which are
huge topics in their own right, so at best I can hope only to, in the
future, throw out more elaborate hints than I am right at this moment.
I mention this missed opportunity because of a historical phenomenon which
keeps impressing me more and more strongly as I learn more about
contemporary solition theory and Lie analysis. In previous posts I have
mentioned the visit to Paris in 1870 of two young mathematicians studying
in Berlin, Felix Klein and Sophus Lie. Among the people they talked to
was Gaston Darboux, the inventor of "Darboux transformations", a kind of
generalization of Baecklund transformations. I have previously tried to
sketch how the crucible of the early ideas of Klein on "Kleinian geometry"
and the early ideas of Lie on Lie analysis of differential equations, Lie
algebras, and Lie groups, was--- algebraic geometry! It is interesting
that soliton theorists seem to currently be uncovering the algebraic roots
of their subject! If I forget to return to this theme in future posts, I
hope someone will remind me, because I think it's important.
Oh dear, once again I'm out of time and so forth, and I never even got to
the Baecklund morphisms for the sine-Gordon equation, much less the KdV!
Oh well, more later, access permitting.
"T. Essel" (hiding somewhere in cyberspace)
Gerard Westendorp
Jul12-04, 03:44 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nGerard Westendorp wrote:\n\n\n[..]\n\n> On the other hand, you could also imagine a 2D phase space in\n> which the solutions are topologically different from concentric\n> curves. Orbits could split and/or rejoin in time.\n\nActually, maybe not actually split or rejoin.\nYou can however, have orbits (=lines traced out by solutions\nin phase space, parametrized by time)\nattracted to a point where orbits get "infinitely closely\ntogether". In other words, solutions from a large region of\nphase space all time-evolve to a point very near the attractor.\n\n\n> In the first case, you can take a point in phase space, and trace\n> it back in time, and find out about its past in a unique way.\n> A kind of "inverse scattering". (I read that the possibility to do\n> inverse scattering is a characteristic of completely integrable\n> systems).\n>\n> In the second case, it is possible that you cannot trace\n> an orbit back uniquely. Because if you trace back to a "split\n> point", you do not know which way to go. In this sense, it is not\n> completely integrable.\n\nHmm, I\'ll have to take that back, I think. > completely integrable\nis apparently something different, I am reading about\nit at the moment. Also, I think you normally do not get points\nin phase which actually split orbits. But you can get the chaos\neffect, in which a very small shift in phase space gives a very\nlarge change later on.\n\nGerard\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Gerard Westendorp wrote:
[..]
> On the other hand, you could also imagine a 2D phase space in
> which the solutions are topologically different from concentric
> curves. Orbits could split and/or rejoin in time.
Actually, maybe not actually split or rejoin.
You can however, have orbits (=lines traced out by solutions
in phase space, parametrized by time)
attracted to a point where orbits get "infinitely closely
together". In other words, solutions from a large region of
phase space all time-evolve to a point very near the attractor.
> In the first case, you can take a point in phase space, and trace
> it back in time, and find out about its past in a unique way.
> A kind of "inverse scattering". (I read that the possibility to do
> inverse scattering is a characteristic of completely integrable
> systems).
>
> In the second case, it is possible that you cannot trace
> an orbit back uniquely. Because if you trace back to a "split
> point", you do not know which way to go. In this sense, it is not
> completely integrable.
Hmm, I'll have to take that back, I think. > completely integrable
is apparently something different, I am reading about
it at the moment. Also, I think you normally do not get points
in phase which actually split orbits. But you can get the chaos
effect, in which a very small shift in phase space gives a very
large change later on.
Gerard
Gerard Westendorp
Jul12-04, 03:44 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\ntessel@tum.bot wrote:\n\n[..]\n\n> Hi again, Oz and Gerard, I have some great web references for you!\n\n\nThanks, I\' ve been reading, and I \'ve learned some interesting stuff.\nBut it is also a bit mind blowing to see how much there is out there.\n\n[..]\n\n> The example of the CR system already illustrates the essential character\n> of Baecklund automorphisms. We have a space of functions which is the\n> space of solutions to some PDE (often second order and often nonlinear---\n> unlike the case here, since Laplace\'s equation is linear), perhaps with\n> some boundary conditions like "rapidly vanishing at spatial infinity".\n> The Baecklund automorphism is a map from this function space to itself,\n> and typically is written as a system of PDEs (often first order and\n> sometimes linear, as here). But this system is inconsistent without a\n> -consistency condition-, which is another PDE (often second order and\n> often nonlinear)--- and this PDE is none other than the PDE we started\n> with when we were defining our function space!\n\n\nThis reminds me a bit a the ladder operators for the quantum oscillator.\nIf you start with the lowest energy mode, you get a wave function that\nis something like\n\npsi = exp(-x^2/2)\n\nThen you can apply the creation operator (x+id/dx)\n(Ignoring all minus signs and normalization factors)\n\nto get something like\n\n\npsi = (x + ix) exp(-x^2/2)\n\nYou can repeat this procedure to get lots of new solutions\nto quantum oscillator from the first one.\n\nAnother similarity with Backlund morphism is that conserved quantities\nare being generated, in this case the energy carried in the eigenmodes,\neach of which is conserved separately.\n\nBut you don\'t have a consistency condition here, as far as I can see.\n\nGerard\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>tessel@tum.bot wrote:
[..]
> Hi again, Oz and Gerard, I have some great web references for you!
Thanks, I' ve been reading, and I 've learned some interesting stuff.
But it is also a bit mind blowing to see how much there is out there.
[..]
> The example of the CR system already illustrates the essential character
> of Baecklund automorphisms. We have a space of functions which is the
> space of solutions to some PDE (often second order and often nonlinear---
> unlike the case here, since Laplace's equation is linear), perhaps with
> some boundary conditions like "rapidly vanishing at spatial infinity".
> The Baecklund automorphism is a map from this function space to itself,
> and typically is written as a system of PDEs (often first order and
> sometimes linear, as here). But this system is inconsistent without a
> -consistency condition-, which is another PDE (often second order and
> often nonlinear)--- and this PDE is none other than the PDE we started
> with when we were defining our function space!
This reminds me a bit a the ladder operators for the quantum oscillator.
If you start with the lowest energy mode, you get a wave function that
is something like
\psi = \exp(-x^2/2)
Then you can apply the creation operator (x+id/dx)
(Ignoring all minus signs and normalization factors)
to get something like
\psi = (x + ix) \exp(-x^2/2)
You can repeat this procedure to get lots of new solutions
to quantum oscillator from the first one.
Another similarity with Backlund morphism is that conserved quantities
are being generated, in this case the energy carried in the eigenmodes,
each of which is conserved separately.
But you don't have a consistency condition here, as far as I can see.
Gerard
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\ntessel@tum.bot writes\n>A few days ago I found some web pages with nice animated gifs showing\n>soliton solutions of the first three, comprising, by universal\n>acclamation, the three "classic" soliton equations:\n>\n>http://www.math.h.kyoto-u.ac.jp/~takasaki/soliton-lab/gallery/solitons/kdv-\n>e.html\n\nThat\'s mindblowing.\n\nHas anyone produced solitons from (1+3)D equations?\n\nIn passing I would like to say that your post, in its enthusiasm and\nrange, are verging on the baezesque.\n\nWhat puzzles me is why research into solitons and their possible use for\nexplaining particles seemed to have lapsed. I would admit that the maths\nmight be arcane (hey most maths seems arcane to me) but that doesn\'t\nseem to have stopped people in the past.\n\nI seem to remember reading (in the distant past) that some solitons at\nleast are constrained to a specific amplitude(s), and that an arbitrary\npulse would \'decompose\' into a selection of stable \'soliton modes\'. Have\nyou come across this in your wideranging travels of the literature?\n\n--\nOz\nThis post is worth absolutely nothing and is probably fallacious.\n\nBTOPENWORLD address about to cease. DEMON address no longer in use.\n>>Use oz@farmeroz.port995.com<<\nozacoohdb@despammed.com still functions.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>tessel@tum.bot writes
>A few days ago I found some web pages with nice animated gifs showing
>soliton solutions of the first three, comprising, by universal
>acclamation, the three "classic" soliton equations:
>
>http://www.math.h.kyoto-u.ac.jp/~takasaki/soliton-lab/gallery/solitons/kdv-
>e.html
That's mindblowing.
Has anyone produced solitons from (1+3)D equations?
In passing I would like to say that your post, in its enthusiasm and
range, are verging on the baezesque.
What puzzles me is why research into solitons and their possible use for
explaining particles seemed to have lapsed. I would admit that the maths
might be arcane (hey most maths seems arcane to me) but that doesn't
seem to have stopped people in the past.
I seem to remember reading (in the distant past) that some solitons at
least are constrained to a specific amplitude(s), and that an arbitrary
pulse would 'decompose' into a selection of stable 'soliton modes'. Have
you come across this in your wideranging travels of the literature?
--
Oz
This post is worth absolutely nothing and is probably fallacious.
BTOPENWORLD address about to cease. DEMON address no longer in use.
>>Use oz@farmeroz.port995.com<<
ozacoohdb@despammed.com still functions.
tessel@tum.bot
Jul13-04, 02:37 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Mon, 12 Jul 2004, Oz wrote:\n\n> >http://www.math.h.kyoto-u.ac.jp/~takasaki/soliton-lab/gallery/solitons/kdv-\n> >e.html\n>\n> That\'s mindblowing.\n\nIndeed it is!\n\n> Has anyone produced solitons from (1+3)D equations?\n\nThe KdV (which allows only right moving waves in one dimension) is a\nspecialization of two very important equations which are a bit more\ncomplicated:\n\nKdV ---> Boussinesq (undo "follow only right moving waves"),\nthe basis one dimensional shallow water wave equation\n|\n|\nV\n\nKadmotsev-Peviashvili (two dimensional KdV)\n\nand there is definitely a two dimensional one allowing for propagation of\nsolitonic "ridges" in any direction on the two dimensional surface of a\nflat shallow body of water. All of these ultimately arise, of course,\nfrom more fundamental equations of hydrodynamics plus uniform\n"gravitational field", subject to appropriate approximations summarized by\nthe phrase "shallow water", so this isn\'t surprising.\n\nInterestingly enough, I have seen some comments to the effect that soliton\nequations seem to like two variables x,t. But certainly the NLS should\nadmit any dimension; in particular, the "self-focusing laser pulses" which\nI mentioned are fully three dimensional:\n\nhttp://www.sfu.ca/~renns/lbullets.html\n\nhttp://www.ma.hw.ac.uk/~chris/dst/\n\nand for unwary Win95 folk, there is this\n\nhttp://www.csa.ru/Inst/bogd_dep/nonlin/vizual.htm\n\n> In passing I would like to say that your post, in its enthusiasm and\n> range, are verging on the baezesque.\n\nActually, it is Palais\'s paper and the book edited by Fordy and Wood which\nis Baezesque. I\'m just trying to summarize a bit of what you will find in\njust these two tiny places.\n\n> What puzzles me is why research into solitons and their possible use for\n> explaining particles seemed to have lapsed.\n\nResearch into -solitons- has NOT lapsed!!!\n\nProbably you know that and just misspoke?\n\nAs for "explaining particles", well, hold that thought, i.e. keep asking\nthis question, since right now I don\'t have a really good answer for you.\n\nBut let\'s try to nail down better exactly what has intrigued you or caused\nyou to insist that there should be some connection with fundamental\nparticle physics here.\n\nProbably most readers who know my style expected me to say "if you knew\nmore about fundamental particle physics you\'d know that because X,Y,Z this\ncan\'t work out". Since I don\'t know any more about particle physics than\nyou do, you are safe from this kind of comment, at least from me. (Not so\nBaezesque after all, no fireballs...)\n\nI think you have been intrigued by the "particle-wave duality" of the KdV\nsolitons which I have briefly described. But that is in quotes, because\nit seems to me that the analogy is vague and not even very close. I\nalready pointed out, for example, that I know of nothing in fundamental\nparticle physics which says "particles of energy 1 (not momentum 1) only\ntravel at speed 1, and particles of energy 2 only travel at speed 2".\nAFAIK, the analogy is mostly just that particle number is conserved in say\nbilliard balls, and in KdV on a circle, the number of solitons running\naround the circle (at various speeds, proportional to their height) is\nconserved. Hard billiard balls are resiliant to collisions, but they\ndon\'t pass through each other or behave in collisions anything like our\nsolitons, as far as I can see. So, any "direct interpretation" in which\nwe try to model particles as KdV solitons with x = location and t = time\nseems to me clearly misguided.\n\nIt -is- however very striking that there is a simple and profoundly\nimportant relation between the KdV and the Schroedinger equation, as Lax\ndiscovered. This is not a vague "analogy" but a precise relationship, so\nthat\'s what impresses -me- here. But as you look into this relationship,\nyou see that for the wave functions psi(x,t) which appear in the Lax\nformulation, we need to think of these as a family, labeled by parameter\nt, of functions in the variable x, not as functions of space and time!\nSo, as often happens, the interpretation, which is what you are really\nconcerned about, is very surprising (and a bit mystifying).\n\n> I would admit that the maths might be arcane (hey most maths seems\n> arcane to me) but that doesn\'t seem to have stopped people in the past.\n>\n> I seem to remember reading (in the distant past) that some solitons at\n> least are constrained to a specific amplitude(s),\n\nYou are probably thinking of the fact that KdV solitons have speed\nproportional to their amplitude (maximal height).\n\n> and that an arbitrary pulse would \'decompose\' into a selection of stable\n> \'soliton modes\'. Have you come across this in your wideranging travels\n> of the literature?\n\nThis is exactly the thing I hope to find time to try to explain tommorrow:\nthe Fermi-Pasta-Ulam numerical experiment. That\'s the picture I already\npointed you at (keep the link--- you\'ll need it again!). FPU started with\na sine wave and evolved by KdV equation\n\nu_t + u u_x + u_(xxx) = 0\n\non a circle (not an infinite line).\n\nWhen u is small, this is approximately\n\nu_t + u_(xxx) = 0\n\nwhich is a linear dispersive wave equation (we used it to discuss\nnonlinear dispersion relations).\n\nWhen u is not small, but u_(xxx) is small, this is approximately the\nnonlinear advection equation\n\nu_t + u u_x = 0\n\nThis turns out to evolve in a graphically vivid manner: start with some\ngraph and move points horizontally to right with speed depending on\nheight. Higher points on the graph move rightwards faster, so no matter\nwhat initial profile you start with (say a sine), your wavefront steepens.\nThis is called "breaking" at the beach! (When surf curls over, the graph\nof the surface of the water becomes three-valued, so it\'s no longer a\nsingle valued function and lotsa math breaks down and cries when that\nhappens.)\n\nSo, looking at that animated gif again, start with sine profile and evolve\nby KdV. At first u_(xxx) is small and this is indistinguishable from\nevolution by the nonlinear advection equation. So, the profile steepens\nuntil it looks much like\n\n/|\n/ |___\n\nJust as it is about to "break", u_(xxx) gets large and intervenes to\nprevent catastrophe. Suddenly, the sawtooth profile splits into a\nprocession of solitons of different heights-- say eight of them.\n\n(If you play around with different profiles, you\'ll find that the number\nof solitions and their spacing and heights depends on the starting profile\nin some nonobvious ways, but there should generally appear some finite\nnumber of solitons.)\n\nSince KdV solitons travel at speeds proportional to height, the smaller\nones (to left of main wavecrest) travel more slowly, i.e. move backward\nwrt this frame where we move with the biggest peak (the tallest\nwavecrest). This is all happening on a -circle-, however, so they wrap\naround and come back from the right hand side.\n\nAs each one collides with the main peak, you have this nonlinear\ninteraction (note the cute "bounce" when this happens) and thereafter the\ntwo solitons reappear unscathed: same height and speed, but with different\nphases, i.e. they are displaced from the position they would occupy if the\ncollision had never happened.\n\nThen each one collides with the next peak at left, etc.\n\nBut, they don\'t split up into zillions of different solitons-- the number\nof solitons is conserved.\n\nNext: recurrence. After a not very long time, the original sine wave\n-reappears-, very nearly. And this is absolutely amazing!\n\nRoughly speaking, we have zillions of oscillators, so from general\nprinciples from ergodic theory we\'d expect a very long recurrence time,\ni.e. we\'d never expect to see the initial sine profile reappear in our\nlifetime. But it does! This was one of the great surprises of FPU.\n\nIt turns out that this amazing recurrence happens because the original\noscillators aren\'t important, only the number of solitons (eight) is\nimportant. And eight phases lining up the same way again after a not very\nlong period of time is not as suprising as 100 zillions phases suddenly\nlining up the same way again.\n\nUnfortunately that page doesn\'t show the recurrence phenomenon, but the\n"picture book" I recommended to you does!\n\n(BTW, I lied above--- FPU set up a one-dimensional lattice consisting of\nfinitely many evenly spaced oscillators on a circle, and evolved an\ninitial vibration state using a certain nearest-neighbor interaction, but\nZabusky and Kruskal showed that in continuum limit this lattice model can\nbe reduced to KdV on a circle).\n\nIf you are not near Snibston, what about Union Canal near Heriot-Watt\nUniversity?\n\nhttp://www.ma.hw.ac.uk/solitons/canal1.html\n\n"T. Essel" (hiding somewhere in cyberspace)\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Mon, 12 Jul 2004, Oz wrote:
> >http://www.math.h.kyoto-u.ac.jp/~takasaki/soliton-lab/gallery/solitons/kdv-
> >e.html
>
> That's mindblowing.
Indeed it is!
> Has anyone produced solitons from (1+3)D equations?
The KdV (which allows only right moving waves in one dimension) is a
specialization of two very important equations which are a bit more
complicated:
KdV ---> Boussinesq (undo "follow only right moving waves"),
the basis one dimensional shallow water wave equation
|
|
V
Kadmotsev-Peviashvili (two dimensional KdV)
and there is definitely a two dimensional one allowing for propagation of
solitonic "ridges" in any direction on the two dimensional surface of a
flat shallow body of water. All of these ultimately arise, of course,
from more fundamental equations of hydrodynamics plus uniform
"gravitational field", subject to appropriate approximations summarized by
the phrase "shallow water", so this isn't surprising.
Interestingly enough, I have seen some comments to the effect that soliton
equations seem to like two variables x,t. But certainly the NLS should
admit any dimension; in particular, the "self-focusing laser pulses" which
I mentioned are fully three dimensional:
http://www.sfu.ca/~renns/lbullets.html
http://www.ma.hw.ac.uk/~chris/dst/
and for unwary Win95 folk, there is this
http://www.csa.ru/Inst/bogd_dep/nonlin/vizual.htm
> In passing I would like to say that your post, in its enthusiasm and
> range, are verging on the baezesque.
Actually, it is Palais's paper and the book edited by Fordy and Wood which
is Baezesque. I'm just trying to summarize a bit of what you will find in
just these two tiny places.
> What puzzles me is why research into solitons and their possible use for
> explaining particles seemed to have lapsed.
Research into -solitons- has NOT lapsed!!!
Probably you know that and just misspoke?
As for "explaining particles", well, hold that thought, i.e. keep asking
this question, since right now I don't have a really good answer for you.
But let's try to nail down better exactly what has intrigued you or caused
you to insist that there should be some connection with fundamental
particle physics here.
Probably most readers who know my style expected me to say "if you knew
more about fundamental particle physics you'd know that because X,Y,Z this
can't work out". Since I don't know any more about particle physics than
you do, you are safe from this kind of comment, at least from me. (Not so
Baezesque after all, no fireballs...)
I think you have been intrigued by the "particle-wave duality" of the KdV
solitons which I have briefly described. But that is in quotes, because
it seems to me that the analogy is vague and not even very close. I
already pointed out, for example, that I know of nothing in fundamental
particle physics which says "particles of energy 1 (not momentum 1) only
travel at speed 1, and particles of energy 2 only travel at speed 2".
AFAIK, the analogy is mostly just that particle number is conserved in say
billiard balls, and in KdV on a circle, the number of solitons running
around the circle (at various speeds, proportional to their height) is
conserved. Hard billiard balls are resiliant to collisions, but they
don't pass through each other or behave in collisions anything like our
solitons, as far as I can see. So, any "direct interpretation" in which
we try to model particles as KdV solitons with x = location and t = time
seems to me clearly misguided.
It -is-[/itex] however very striking that there is a simple and profoundly
important relation between the KdV and the Schroedinger equation, as Lax
discovered. This is not a vague "analogy" but a precise relationship, so
that's what impresses -me- here. But as you look into this relationship,
you see that for the wave functions \psi(x,t) which appear in the Lax
formulation, we need to think of these as a family, labeled by parameter
t, of functions in the variable x, not as functions of space and time!
So, as often happens, the interpretation, which is what you are really
concerned about, is very surprising (and a bit mystifying).
> I would admit that the maths might be arcane (hey most maths seems
> arcane to me) but that doesn't seem to have stopped people in the past.
>
> I seem to remember reading (in the distant past) that some solitons at
> least are constrained to a specific amplitude(s),
You are probably thinking of the fact that KdV solitons have speed
proportional to their amplitude (maximal height).
> and that an arbitrary pulse would 'decompose' into a selection of stable
> 'soliton modes'. Have you come across this in your wideranging travels
> of the literature?
This is exactly the thing I hope to find time to try to explain tommorrow:
the Fermi-Pasta-Ulam numerical experiment. That's the picture I already
pointed you at (keep the link--- you'll need it again!). FPU started with
a sine wave and evolved by KdV equation
[itex]u_t + u u_x + u_(xxx) =
on a circle (not an infinite line).
When u is small, this is approximately
u_t + u_(xxx) =
which is a linear dispersive wave equation (we used it to discuss
nonlinear dispersion relations).
When u is not small, but u_(xxx) is small, this is approximately the
nonlinear advection equation
u_t + u u_x =
This turns out to evolve in a graphically vivid manner: start with some
graph and move points horizontally to right with speed depending on
height. Higher points on the graph move rightwards faster, so no matter
what initial profile you start with (say a sine), your wavefront steepens.
This is called "breaking" at the beach! (When surf curls over, the graph
of the surface of the water becomes three-valued, so it's no longer a
single valued function and lotsa math breaks down and cries when that
happens.)
So, looking at that animated gif again, start with sine profile and evolve
by KdV. At first u_(xxx) is small and this is indistinguishable from
evolution by the nonlinear advection equation. So, the profile steepens
until it looks much like
/|
/ |___
Just as it is about to "break", u_(xxx) gets large and intervenes to
prevent catastrophe. Suddenly, the sawtooth profile splits into a
procession of solitons of different heights-- say eight of them.
(If you play around with different profiles, you'll find that the number
of solitions and their spacing and heights depends on the starting profile
in some nonobvious ways, but there should generally appear some finite
number of solitons.)
Since KdV solitons travel at speeds proportional to height, the smaller
ones (to left of main wavecrest) travel more slowly, i.e. move backward
wrt this frame where we move with the biggest peak (the tallest
wavecrest). This is all happening on a -circle-, however, so they wrap
around and come back from the right hand side.
As each one collides with the main peak, you have this nonlinear
interaction (note the cute "bounce" when this happens) and thereafter the
two solitons reappear unscathed: same height and speed, but with different
phases, i.e. they are displaced from the position they would occupy if the
collision had never happened.
Then each one collides with the next peak at left, etc.
But, they don't split up into zillions of different solitons-- the number
of solitons is conserved.
Next: recurrence. After a not very long time, the original sine wave
-reappears-, very nearly. And this is absolutely amazing!
Roughly speaking, we have zillions of oscillators, so from general
principles from ergodic theory we'd expect a very long recurrence time,
i.e. we'd never expect to see the initial sine profile reappear in our
lifetime. But it does! This was one of the great surprises of FPU.
It turns out that this amazing recurrence happens because the original
oscillators aren't important, only the number of solitons (eight) is
important. And eight phases lining up the same way again after a not very
long period of time is not as suprising as 100 zillions phases suddenly
lining up the same way again.
Unfortunately that page doesn't show the recurrence phenomenon, but the
"picture book" I recommended to you does!
(BTW, I lied above--- FPU set up a one-dimensional lattice consisting of
finitely many evenly spaced oscillators on a circle, and evolved an
initial vibration state using a certain nearest-neighbor interaction, but
Zabusky and Kruskal showed that in continuum limit this lattice model can
be reduced to KdV on a circle).
If you are not near Snibston, what about Union Canal near Heriot-Watt
University?
http://www.ma.hw.ac.uk/solitons/canal1.html
"T. Essel" (hiding somewhere in cyberspace)
tessel@tum.bot
Jul13-04, 02:37 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Mon, 12 Jul 2004, Gerard Westendorp wrote:\n\n> Gerard Westendorp wrote:\n>\n> > (I read that the possibility to do\n> > inverse scattering is a characteristic of completely integrable\n> > systems).\n\nThis is correct; did I say otherwise earlier?--- if so, I know better now\nand I emphatically correct myself :-/\n\n> > In the second case, it is possible that you cannot trace\n> > an orbit back uniquely. Because if you trace back to a "split\n> > point", you do not know which way to go. In this sense, it is not\n> > completely integrable.\n>\n> Hmm, I\'ll have to take that back, I think. > completely integrable\n> is apparently something different, I am reading about\n> it at the moment. Also, I think you normally do not get points\n> in phase which actually split orbits. But you can get the chaos\n> effect, in which a very small shift in phase space gives a very\n> large change later on.\n\nI don\'t understand what you mean by "split orbits" here. (Some kind of\nbifurcation?)\n\nI\'m not going to have time to say very much today, unfortunately, but let\nme try again to give a rough characterization of "soliton equation":\n\n* true solitons are special solitary waves (one wavecrest) which -do-\nexhibit quasiperiodicity, recurrence (much faster than expected from\nergodic theory), persistence, speed depending (linearly, at leaast for KdV\nsolitons) upon amplitude, but -don\'t- exhibit dispersion or dissipation of\nthe wavecrest,\n\n* when true true solitons collide, they -do- reconstistute themselves\n"unaltered" (same speed, amplitude, sense) except for phase; they -do not-\n"radiate away energy" upon collision,\n\n* the unusual stability of solitons is explained by the existence of an\ninfinite hierarchy of -independent- conservation laws for the parent PDE;\ncompare ordinary Hamiltonian systems, which admit only finitely many\n-independent- conservation laws (e.g. the Hamiltonian itself is\nconserved),\n\n* that is, a solution u(x,t) is characterized by an infinite list of\nquantities which are invariant during the evolution of the solution\nu(x,t),\n\n* this infinite hierarchy is very closely related to the existence of an\ninfinite dimensional group of Lie-Baecklund symmetries which are "hidden\nsymmetries" from the POV of Lie analysis,\n\n* there may be a "Baecklund automorphism" on the solution space, and a\n"Baecklund morphism" to/from the solution space of another equation, these\nmorphisms respect conserved quantities,\n\n* "soliton equations" (PDEs admitting soliton solutions and having\nproperties as above) seem to generally arise as "continuum limits" of\ninteresting discrete models (this should be of particular interest to you,\nGerard!), often in highly unexpected ways,\n\n* a given soliton equation typically models an astonishing variety of\nthings, e.g. cubic NLS apparently models "self-focusing" laser pulses\n(happens in suitable materials for very high frequency monochromatic\nwaves, not be confused with "internal reflection" of ordinary fiber optic\ncommunication, AFAIK) which telecommunications is trying to adapt to\nincrease the capacity of transatlantic fiber optic cables by a factor of a\nhundred (i.e. to near the theoretical optimum), but also describes in -an\nentirely different way- the surface swept out in E^3 by an vortex filament\n(see below),\n\n* soliton equations admit a "Lax formulation", in which we write down an\noverdetermined system of equations whose "compatibility condition" is the\noriginal PDE; this formulation introduces:\n\n(a) a self-adjoint operator L on a suitable Hilbert space (think of\nbounded wave functions psi), where L "has a good scattering theory"; for\nthe KdV, L is just the time-independent Schroedinger operator\n\nL = @_(xx) + u\n\nwhere u(x,t) now acts as the potential of our Schroedinger operator, with\nx the variable and t now a parameter; as t varies the potential evolves,\n\n(b) an operator B, carefully chosen such that the Lax equation L_t = [B,L]\nimplies the original PDE; for KdV we take\n\nB = -4 @_(xxx) + 3 (u @_x + @_x u)\n\nand Lax showed how to also recover sine-Gordon, etc., in this way (i.e. as\nthe compatibility condition for a suitable Lax system),\n\n* actually, we get an infinite hierarchy of Lax systems (and associated\nPDEs) this way from our initial one; in particular, an infinite sequence\nof "higher KdV" or "higher sine-Gordon" equations,\n\n* this Lax formulation secretly involves a connection omega in a tangent\nbundle TG, where G is some Lie group such as SL(2,R); IOW, omega is a\nmatrix valued one-form\n\nomega = A dx + B dt\n\nwhere A is something like A = @_x - L, and where the covariant derivatives\nare\n\nD_x = @_x - A\n\nD_t = @_t - B\n\n* the Lax equation now takes the form\n\nA_t - B_x - [A,B] = 0\n\nwhich says that the connection has -zero curvature- (!), and this is the\ncompatibility condition for the (overdetermined) Lax system\n\npsi_x = A psi\n\npsi_t = B psi\n\n* the self-adjoint operators L(t) given by the Lax system are all\nunitarily conjugate to one another, hence "unitary evolution", the\noperators B(t) are skew-adjoint,\n\n* this "connection formulation" of the Lax system is secretly a\ngeneralization of the situation in classical surface theory (differential\ngeometry), where the Codazzi equation is the compability condition for the\nWeingarten equation; thus for example\n\n(a) solutions of the sine-Gordon equation\n\nu_(pq) = sin(u)\n\ncorrespond to isometry classes in E^3 of surfaces locally isometric to\nH^2 (in the null chart or "characteristic form" written above, these are\nequipped with a coordinate chart in which the curves p = p0, q = q0 are\n"asymptotic lines" in the sense of classical surface theory [in a region\nwith negative Gaussian curvature]; in the Cartesian form, they are\nequipped with a coordinate chart in which the curves p=p0, q=q0 are "lines\nof principle curvature" in the sense of classical surface theory),\n\n(b) solutions of the E^2 Liouville equation\n\nu_(xx) + u_(yy) = exp(u)\n\ncorrespond to isometry classes in E^3 of minimal surfaces (vanishing mean\ncuvature, i.e. traceless extrinsic curvature tensor),\n\n(c) solutions of Toda\'s system correspond to isometry classes of\nWillmore tori, which have constant trace extrinsic curvature,\n\n(d) solutions of the "cubic NLS" correspond to isometry classes of\nHasimoto surfaces, which are the trace of the evolution over time of a\n-vortex filament-, which geometrically can be described as a closed space\ncurve which evolves by moving (at point P on the curve) in the direction\nof its -binormal- vector at P (Frenet-Serret frame) with a speed given by\nits -torsion- at P,\n\n(wow, huh?!!!!)\n\n(e) etc., including Willmore tori, which are beautiful generalizations\nof the classical Hopf map SU(2) --> S^2; c.f. my misunderstood comments in\nthe spin thread,\n\n* these geometric realizations of solitons as isometric immersions of\n"constant curvature" surfaces are often quite literally -beautiful- to\nbehold, as in artistic; if classical geometers like Darboux (c.f. again\nthe visit of Lie and Klein to Paris in 1870) and Bianchi (see the\nillustration in the book edited by Ford and Wood of an n-soliton solution\nof the sine-Gordon equation as an isometrically immersed surface, which he\ncalled an "n-bubble") had only had even primitive digital computers, a\nmuch wider audience than a handful of inititiates would have quickly\nappreciated the beauty of say Dobriner\'s surfaces (isometric immersions of\nvarious constant curvature surfaces in E^3, in which one of the two\nfamilies of principle curvatures appears as a family of -circles- in E^3;\nthis family usually twists and turns, of course),\n\n* if only they had considered something like u_(pq) as a -wave equation-,\nthey would have discovered the striking properties of collisions of\n"kinks", and probably would have discovered IST etc. fifty or a hundred\nyears earlier than in fact happened!\n\n* the spectral parameter lambda in the Lax formulation (which I somehow\nomitted above--- there\'s a lot to say about the Lax formulation in its\nvarious guises!) corresponds to a kind of "deformation parameter" for such\nclassical family of immersed surfaces, and a consequence of Lax\'s\nformulation is the fact that lambda_t = 0, hence "isospectral flow",\n\n* c.f. Hamiltonian systems and "integrable systems", the existence of the\ninfinite hierarchy of conservation laws (previously used to "explain" the\nstability of soliton solutions), is "explained" in terms of an infinite\ndimensional group of "hidden" symplectic automorphisms,\n\n* this automorphism group is secretly a -loop group-, i.e. a space of\nmappings\n\nS^1 --> G\n\nwhere G is our Lie group mentioned above (connection in TG), and then the\nscattering transformation and its inverse map between a subgroup of the\nloop group and "the space of regular potentials" (or something like this,\nI think),\n\n* in sum, the story of solitons is full of absolutely amazing turns of\nplot!\n\nComing back to one of our questions: Are there many "soliton equations",\nor are the KdV and friends somehow incredibly special? The answer seems\nto be -"both"-; that is, KdV and friends are somehow "universal", so they\nwill arise at least as good approximations in many "near equilibrium"\nsituations, especially in the context of Hamiltonian dynamics; this covers\nan awful lot of territory, so many soliton equations are known, but they\nshould all somehow really come down to the KdV at heart, or something like\nthis, I think.\n\nThe book edited by Ford & Wood, and the paper by Palais, which I cited\nlast time, are both absolutely -superb- sources of information. As you\ncan probably tell, I am still absorbing what they say! I\'d like to try to\nsummarize what they say in a much more organized fashion when I get a\nchance. But first (not today) I need to say more about how to use IST to\nsolve the nonlinear Cauchy problem\n\ngiven u(x,0), find u(x,t) such that\n\nu_t + u u_x + u_(xxx) = 0\n\nand to compare this with the usual -Fourier transform- for the linear\nCauchy problem:\n\ngiven u(x,0), find u(x,t) such that\n\nu_t + u_(xxx) =0\n\nand to explain "wavefront steepening by nonlinear advection" for\n\nu_t + u u_x = 0\n\n(where we have KdV and two different "limits" of same).\n\nCaveat: the above should be regarded as a hysterical outpouring of joy\nrather than the result of mature understanding :-/ (Joy, because I am\ndiscovering that my past insistence upon the value of Cartan\'s viewpoint\nis amply justified in soliton theory.) If the recent past is any guide, I\nmay have to amend some of what I just said when I understand this stuff\nbetter. But one thing is clear: hysterical transports of joy is probably\nnot an unsuitable initial reaction to this extraordinarily beautiful\nsubject.\n\n"T. Essel" (hiding somewhere in cyberspace)\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Mon, 12 Jul 2004, Gerard Westendorp wrote:
> Gerard Westendorp wrote:
>
> > (I read that the possibility to do
> > inverse scattering is a characteristic of completely integrable
> > systems).
This is correct; did I say otherwise earlier?--- if so, I know better now
and I emphatically correct myself :-/
> > In the second case, it is possible that you cannot trace
> > an orbit back uniquely. Because if you trace back to a "split
> > point", you do not know which way to go. In this sense, it is not
> > completely integrable.
>
> Hmm, I'll have to take that back, I think. > completely integrable
> is apparently something different, I am reading about
> it at the moment. Also, I think you normally do not get points
> in phase which actually split orbits. But you can get the chaos
> effect, in which a very small shift in phase space gives a very
> large change later on.
I don't understand what you mean by "split orbits" here. (Some kind of
bifurcation?)
I'm not going to have time to say very much today, unfortunately, but let
me try again to give a rough characterization of "soliton equation":
* true solitons are special solitary waves (one wavecrest) which -do-
exhibit quasiperiodicity, recurrence (much faster than expected from
ergodic theory), persistence, speed depending (linearly, at leaast for KdV
solitons) upon amplitude, but -don't- exhibit dispersion or dissipation of
the wavecrest,
* when true true solitons collide, they -do- reconstistute themselves
"unaltered" (same speed, amplitude, sense) except for phase; they -do not-
"radiate away energy" upon collision,
* the unusual stability of solitons is explained by the existence of an
infinite hierarchy of -independent- conservation laws for the parent PDE;
compare ordinary Hamiltonian systems, which admit only finitely many
-independent- conservation laws (e.g. the Hamiltonian itself is
conserved),
* that is, a solution u(x,t) is characterized by an infinite list of
quantities which are invariant during the evolution of the solution
u(x,t),
* this infinite hierarchy is very closely related to the existence of an
infinite dimensional group of Lie-Baecklund symmetries which are "hidden
symmetries" from the POV of Lie analysis,
* there may be a "Baecklund automorphism" on the solution space, and a
"Baecklund morphism" to/from the solution space of another equation, these
morphisms respect conserved quantities,
* "soliton equations" (PDEs admitting soliton solutions and having
properties as above) seem to generally arise as "continuum limits" of
interesting discrete models (this should be of particular interest to you,
Gerard!), often in highly unexpected ways,
* a given soliton equation typically models an astonishing variety of
things, e.g. cubic NLS apparently models "self-focusing" laser pulses
(happens in suitable materials for very high frequency monochromatic
waves, not be confused with "internal reflection" of ordinary fiber optic
communication, AFAIK) which telecommunications is trying to adapt to
increase the capacity of transatlantic fiber optic cables by a factor of a
hundred (i.e. to near the theoretical optimum), but also describes in -an
entirely different way- the surface swept out in E^3 by an vortex filament
(see below),
* soliton equations admit a "Lax formulation", in which we write down an
overdetermined system of equations whose "compatibility condition" is the
original PDE; this formulation introduces:
(a) a self-adjoint operator L on a suitable Hilbert space (think of
bounded wave functions \psi), where L "has a good scattering theory"; for
the KdV, L is just the time-independent Schroedinger operator
L = @_(xx) + u
where u(x,t) now acts as the potential of our Schroedinger operator, with
x the variable and t now a parameter; as t varies the potential evolves,
(b) an operator B, carefully chosen such that the Lax equation L_t = [B,L]
implies the original PDE; for KdV we take
B = -4 @_(xxx) + 3 (u @_x + @_x u)
and Lax showed how to also recover sine-Gordon, etc., in this way (i.e. as
the compatibility condition for a suitable Lax system),
* actually, we get an infinite hierarchy of Lax systems (and associated
PDEs) this way from our initial one; in particular, an infinite sequence
of "higher KdV" or "higher sine-Gordon" equations,
* this Lax formulation secretly involves a connection \omega in a tangent
bundle TG, where G is some Lie group such as SL(2,R); IOW, \omega is a
matrix valued one-form
\omega = A dx + B dt
where A is something like A = @_x - L, and where the covariant derivatives
are
D_x = @_x - AD_t = @_t - B
* the Lax equation now takes the form
A_t - B_x -[/itex] [A,B] =
which says that the connection has -zero curvature- (!), and this is the
compatibility condition for the (overdetermined) Lax system
\psi_x = A \psi\psi_t = B \psi
* the self-adjoint operators L(t) given by the Lax system are all
unitarily conjugate to one another, hence "unitary evolution", the
operators B(t) are skew-adjoint,
* this "connection formulation" of the Lax system is secretly a
generalization of the situation in classical surface theory (differential
geometry), where the Codazzi equation is the compability condition for the
Weingarten equation; thus for example
(a) solutions of the sine-Gordon equation
u_(pq) = sin(u)
correspond to isometry classes in E^3 of surfaces locally isometric to
H^2 (in the null chart or "characteristic form" written above, these are
equipped with a coordinate chart in which the curves p = p0, q = q0 are
"asymptotic lines" in the sense of classical surface theory [in a region
with negative Gaussian curvature]; in the Cartesian form, they are
equipped with a coordinate chart in which the curves p=p0, q=q0 are "lines
of principle curvature" in the sense of classical surface theory),
(b) solutions of the E^2 Liouville equation
u_(xx) + u_(yy) = \exp(u)
correspond to isometry classes in E^3 of minimal surfaces (vanishing mean
cuvature, i.e. traceless extrinsic curvature tensor),
(c) solutions of Toda's system correspond to isometry classes of
Willmore tori, which have constant trace extrinsic curvature,
(d) solutions of the "cubic NLS" correspond to isometry classes of
Hasimoto surfaces, which are the trace of the evolution over time of a
-vortex filament-, which geometrically can be described as a closed space
curve which evolves by moving (at point P on the curve) in the direction
of its -binormal- vector at P (Frenet-Serret frame) with a speed given by
its -torsion- at P,
(wow, huh?!!!!)
(e) etc., including Willmore tori, which are beautiful generalizations
of the classical Hopf map SU(2) --> S^2; c.f. my misunderstood comments in
the spin thread,
* these geometric realizations of solitons as isometric immersions of
"constant curvature" surfaces are often quite literally -beautiful- to
behold, as in artistic; if classical geometers like Darboux (c.f. again
the visit of Lie and Klein to Paris in 1870) and Bianchi (see the
illustration in the book edited by Ford and Wood of an n-soliton solution
of the sine-Gordon equation as an isometrically immersed surface, which he
called an "n-bubble") had only had even primitive digital computers, a
much wider audience than a handful of inititiates would have quickly
appreciated the beauty of say Dobriner's surfaces (isometric immersions of
various constant curvature surfaces in E^3, in which one of the two
families of principle curvatures appears as a family of -circles- in E^3;
this family usually twists and turns, of course),
* if only they had considered something like u_(pq) as a -wave equation-,
they would have discovered the striking properties of collisions of
"kinks", and probably would have discovered IST etc. fifty or a hundred
years earlier than in fact happened!
* the spectral parameter \lambda in the Lax formulation (which I somehow
omitted above--- there's a lot to say about the Lax formulation in its
various guises!) corresponds to a kind of "deformation parameter" for such
classical family of immersed surfaces, and a consequence of Lax's
formulation is the fact that \lambda_t = 0, hence "isospectral flow",
* c.f. Hamiltonian systems and "integrable systems", the existence of the
infinite hierarchy of conservation laws (previously used to "explain" the
stability of soliton solutions), is "explained" in terms of an infinite
dimensional group of "hidden" symplectic automorphisms,
* this automorphism group is secretly a -loop group-, i.e. a space of
mappings
S^1 --> G
where G is our Lie group mentioned above (connection in TG), and then the
scattering transformation and its inverse map between a subgroup of the
loop group and "the space of regular potentials" (or something like this,
I think),
* in sum, the story of solitons is full of absolutely amazing turns of
plot!
Coming back to one of our questions: Are there many "soliton equations",
or are the KdV and friends somehow incredibly special? The answer seems
to be -"both"-; that is, KdV and friends are somehow "universal", so they
will arise at least as good approximations in many "near equilibrium"
situations, especially in the context of Hamiltonian dynamics; this covers
an awful lot of territory, so many soliton equations are known, but they
should all somehow really come down to the KdV at heart, or something like
this, I think.
The book edited by Ford & Wood, and the paper by Palais, which I cited
last time, are both absolutely -superb- sources of information. As you
can probably tell, I am still absorbing what they say! I'd like to try to
summarize what they say in a much more organized fashion when I get a
chance. But first (not today) I need to say more about how to use IST to
solve the nonlinear Cauchy problem
given u(x,0), find u(x,t) such that
[itex]u_t + u u_x + u_(xxx) =
and to compare this with the usual -Fourier transform- for the linear
Cauchy problem:
given u(x,0), find u(x,t) such that
u_t + u_(xxx) =0
and to explain "wavefront steepening by nonlinear advection" for
u_t + u u_x =
(where we have KdV and two different "limits" of same).
Caveat: the above should be regarded as a hysterical outpouring of joy
rather than the result of mature understanding :-/ (Joy, because I am
discovering that my past insistence upon the value of Cartan's viewpoint
is amply justified in soliton theory.) If the recent past is any guide, I
may have to amend some of what I just said when I understand this stuff
better. But one thing is clear: hysterical transports of joy is probably
not an unsuitable initial reaction to this extraordinarily beautiful
subject.
"T. Essel" (hiding somewhere in cyberspace)
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nOz <oz@farmeroz.port995.com> writes\n>tessel@tum.bot writes\n>>\n>>http://www.math.h.kyoto-u.ac.jp/~takasaki/soliton-lab/gallery/solitons/kdv-\n>>e.html\n>\n>That\'s mindblowing.\n\nhttp://homepages.tversu.ru/~s000154/collision/main.html\n\nAre just wonderful. I particularly like the effect of amplitude, and the\n\'low amplitude\' waves, which do look awfully like a wavebundle, must\nsurely be the sort of thing a particle should be like.\n\nObviously work on solitons hasn\'t stopped, it just hasn\'t got anywhere.\n\nI\'m so excited by it! <swoons>\n\n--\nOz\nThis post is worth absolutely nothing and is probably fallacious.\n\nBTOPENWORLD address about to cease. DEMON address no longer in use.\n>>Use oz@farmeroz.port995.com<<\nozacoohdb@despammed.com still functions.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Oz <oz@farmeroz.port995.com> writes
>tessel@tum.bot writes
>>
>>http://www.math.h.kyoto-u.ac.jp/~takasaki/soliton-lab/gallery/solitons/kdv-
>>e.html
>
>That's mindblowing.
http://homepages.tversu.ru/~s000154/collision/main.html
Are just wonderful. I particularly like the effect of amplitude, and the
'low amplitude' waves, which do look awfully like a wavebundle, must
surely be the sort of thing a particle should be like.
Obviously work on solitons hasn't stopped, it just hasn't got anywhere.
I'm so excited by it! <swoons>
--
Oz
This post is worth absolutely nothing and is probably fallacious.
BTOPENWORLD address about to cease. DEMON address no longer in use.
>>Use oz@farmeroz.port995.com<<
ozacoohdb@despammed.com still functions.
Robert Shaw
Jul13-04, 04:18 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n<tessel@tum.bot> wrote in\n>\n> * c.f. Hamiltonian systems and "integrable systems", the existence of\n> the infinite hierarchy of conservation laws (previously used to\n> "explain" the stability of soliton solutions), is "explained" in terms\n> of an infinite dimensional group of "hidden" symplectic\n> automorphisms,\n>\n> * this automorphism group is secretly a -loop group-, i.e. a space of\n> mappings\n>\n> S^1 --> G\n>\n> where G is our Lie group mentioned above (connection in TG), and then\n> the scattering transformation and its inverse map between a subgroup\n> of the loop group and "the space of regular potentials" (or something\n> like this, I think),\n>\n> * in sum, the story of solitons is full of absolutely amazing turns of\n> plot!\n>\n> Coming back to one of our questions: Are there many "soliton\n> equations", or are the KdV and friends somehow incredibly special?\n\nIs it possible to start from a suitable infinite group and deduce a set\nof PDEs with that \'hidden\' automorphism group, presumably with\nsoliton behaviour.\n\nI\'d assume all such PDEs would be equivalent in some useful sense, maybe\nunder Backland transforms.\n\nIf we started from a different group to the KdV\'s, would we get a\ndifferent set of PDEs, or can only that one group be the \'hidden\'\ninfinite-dimensional symmetry group of a PDE?\n\n\n--\nMatter is fundamentally lazy:- It always takes the path of least effort\nMatter is fundamentally stupid:- It tries every other path first.\nThat is the heart of physics - The rest is details.- Robert Shaw\n\n\n\n\n\n\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky><tessel@tum.bot> wrote in
>
> * c.f. Hamiltonian systems and "integrable systems", the existence of
> the infinite hierarchy of conservation laws (previously used to
> "explain" the stability of soliton solutions), is "explained" in terms
> of an infinite dimensional group of "hidden" symplectic
> automorphisms,
>
> * this automorphism group is secretly a -loop group-, i.e. a space of
> mappings
>
> S^1 --> G
>
> where G is our Lie group mentioned above (connection in TG), and then
> the scattering transformation and its inverse map between a subgroup
> of the loop group and "the space of regular potentials" (or something
> like this, I think),
>
> * in sum, the story of solitons is full of absolutely amazing turns of
> plot!
>
> Coming back to one of our questions: Are there many "soliton
> equations", or are the KdV and friends somehow incredibly special?
Is it possible to start from a suitable infinite group and deduce a set
of PDEs with that 'hidden' automorphism group, presumably with
soliton behaviour.
I'd assume all such PDEs would be equivalent in some useful sense, maybe
under Backland transforms.
If we started from a different group to the KdV's, would we get a
different set of PDEs, or can only that one group be the 'hidden'
infinite-dimensional symmetry group of a PDE?
--
Matter is fundamentally lazy:- It always takes the path of least effort
Matter is fundamentally stupid:- It tries every other path first.
That is the heart of physics - The rest is details.- Robert Shaw
Roger Beresford
Jul14-04, 02:52 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nOz <oz@farmeroz.port995.com> wrote in message <snip}\n> Obviously work on solitons hasn\'t stopped, it just hasn\'t got anywhere.\n<snip>\nYou might like this (fictional rather than scientific) quote from\np167 of "Schild\'s Ladder" by Greg Egan (Gollancz 2001), referring to\nsolitonic fundamental particles:-\n"The right Gaussian [wavepacket] though, in the right environment, was\nthe perfect compromise between uncertainty in position and momentum,\nallowing the shape of the wave to remain unchanged as it moved."\nEgan implies that solitonic wave packets have Heisenberg uncertainty\nand Planck size. He does not mention the other features that make\nsolitons so appealing as particle paradigms - that they can carry\nmodulation (deBroglie waves) and that they pass each other with\ndisplacement of their trajectories (mutual attraction or repulsion).\nAttraction takes place (for KdV solitons) in an intermediate region\nthat is lower and broader than either of the well-separated packets.\nThere are no recognizable "virtual messenger particles" to cause this\nattraction - its all in the interaction between the terms of the\nequation. Missing features (for simple solitons) are combination and\nspontaneous decay, changing the nature of the particles. On the basis\nthat phenomena are governed by equations, there ought to be a\nsophisticated non-linear equation that has such features.\nSolitonic PDE\'s are expressible in "conservative" forms, equating\nthe sums of differentiated (conserved) terms to zero. I use the common\nconvention that Ax & [F(A)]x means differentiation by "x", so n^2 At\n+Axxx +(1+n)(2+n) A^(2/n) Ax =0 (Zabulsky 1963 with n replaced by 1/n\n) gives the KdV and MKdV equations when n=1 or 2. The KdV, MKdV, and\nKP (Kadometsev- Petviashvili) equations include terms Axxx & Ayy,\nwhere "A" is a function of "t" ,"x", & (for KP) "y". They have\ninfinite sets of conserved properties characterized by the parameters\nj or k, because [[A^(i-k) [A^k]x/k]x]x becomes Axxx and [A^(1-j)\nAy/j]y becomes Ayy after differentiation.\nThe linear Schroedinger equation has a conservative form, [2 i M A]t\n+ [hbar Ax]x= 0. Remoissenet ("Waves called Solitons", p29) shows that\nit has modulated travelling Gaussian pulse solutions, but these\ndisperse with time - it describes "information about particles", and\nnot the stable particles themselves. A non-linear term, as in KdV-type\nequations, may counteract the dispersion that arises because wave\nvelocities vary with frequency, to give a particle equation.\nAn alternative approach might use "Unit velocity equations" (related\nto Sum[Ax^2/A]x=0, which differentiates to Sum[A Axx-Ax^2]=0, with\ntime included in the summation variables), which allow non-dispersing\npulses because velocity is independent of frequency. These could have\nsome of the velocity in "rolled-up" (Kaluza-Klein) dimensions for\nparticles with mass, the remaining velocity being in space dimensions;\nmassless particles moving at the speed of light. I have made\nnegligible progress with this concept, despite experimenting with\nnon-linear equations (using Mathematica) for many years, looking for\nstable compact travelling multi-dimensional multi-field pulses.\nA broadened KdV equation can be written (in conservative form) as\n[n^2(r+m) A^(1+m)]t + [ [A^(i-k) [A^k]x/k]x + n(1+nr)A^(1+2/n)\nAx+(r-1)Ax^2/A]x = 0; the KP term [A^(1-j) Ay/j]y can be included.\nThis has all the standard Sech^n solitonic solutions, and their\nrepeating lattice JacobiCN^n solutions when r=1, m=0, n=1 or 2.\nRescaling is needed when n=-1 or -2. This gives an equation solved\nby a rational pulse equation\nA = (a/(1+ (a*(kx*(x+xi-kx*t)^2+ky*(y+yi-ky*t)^2))/2))^n\nwhen it is rescaled to\n(1+n)At A^n - Axxx - Ayyy +(1+1/n)(2(Axx Ax+Ayy Ay)\n-(Ax^3+Ay^3)/A)/A=0,\nand a modulated Gaussian pulse\nA=E^(ao-(jx*(ax+t*jx^2-x))^2-(jy*(ay+t*jy^2-y))^2+\ni*(-(jx^4+jy^4)*t +jx^2*x+jy^2*y))\nwhen it is rescaled to\n4At-Axxx-Ayyy+(Ax Axx+AyAyy)/A=0.\nThere is a repeating lattice Sin[kx-kx^3 t]^p solution which becomes a\ngaussian pulse when p->infinity, analogous to the JacobiCN pulses\nbecoming Sech pulses.\nIn each case, any number of space-like directions X={x,y,z..}can be\nincluded. There may be stable solitons for very specific combinations\nof parameters, but I have NOT been able to find them. When I first\ndiscovered these different pulse shapes, I conjectured that rational\npulses (with square-law decay) might be the massive hadrons, gaussian\npulses the massless photons & neutrinos etc., and sech pulses the\npoint-like low-mass electrons etc..\nWhen i="ii"th root of unity (not just the fourth root), with ii>2,\nthe modulated Gaussian equations have all factors (>2) of "ii" as\npossible phasalities. This introduces field directions in addition to\nthe txyz dimensions. "ii"=12 allows 3, 6, & 12 phases as "Quark-wave"\nhadrons, and 4-phase leptons.\nSome of this is demonstrated in section 5 of HoopGlossary, in\nhttp://Library.wolfram.com/infocenter/Mathsource/4894 (with more to\ncome in the next revision).\nMy hope is that a young iconoclast will take the concepts farther than\nI can. As you rightly state, "it just hasn\'t got anywhere".\nRoger Beresford.\n"No-one can resist the invasion of ideas." (Victor Hugo.)\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Oz <oz@farmeroz.port995.com> wrote in message <snip}
> Obviously work on solitons hasn't stopped, it just hasn't got anywhere.
<snip>
You might like this (fictional rather than scientific) quote from
p167 of "Schild's Ladder" by Greg Egan (Gollancz 2001), referring to
solitonic fundamental particles:-
"The right Gaussian [wavepacket] though, in the right environment, was
the perfect compromise between uncertainty in position and momentum,
allowing the shape of the wave to remain unchanged as it moved."
Egan implies that solitonic wave packets have Heisenberg uncertainty
and Planck size. He does not mention the other features that make
solitons so appealing as particle paradigms - that they can carry
modulation (deBroglie waves) and that they pass each other with
displacement of their trajectories (mutual attraction or repulsion).
Attraction takes place (for KdV solitons) in an intermediate region
that is lower and broader than either of the well-separated packets.
There are no recognizable "virtual messenger particles" to cause this
attraction - its all in the interaction between the terms of the
equation. Missing features (for simple solitons) are combination and
spontaneous decay, changing the nature of the particles. On the basis
that phenomena are governed by equations, there ought to be a
sophisticated non-linear equation that has such features.
Solitonic PDE's are expressible in "conservative" forms, equating
the sums of differentiated (conserved) terms to zero. I use the common
convention that Ax & [F(A)]x means differentiation by "x", so n^2 At+Axxx +(1+n)(2+n) A^(2/n) Ax =0 (Zabulsky 1963 with n replaced by 1/n
) gives the KdV and MKdV equations when n=1 or 2. The KdV, MKdV, and
KP (Kadometsev- Petviashvili) equations include terms Axxx & Ayy,
where "A" is a function of "t" ,"x", & (for KP) "y". They have
infinite sets of conserved properties characterized by the parameters
j or k, because [[A^(i-k) [A^k]x/k]x]x becomes Axxx and [A^(1-j)Ay/j]y becomes Ayy after differentiation.
The linear Schroedinger equation has a conservative form, [2 i M A]t
+ [\hbar Ax]x= . Remoissenet ("Waves called Solitons", p29) shows that
it has modulated travelling Gaussian pulse solutions, but these
disperse with time - it describes "information about particles", and
not the stable particles themselves. A non-linear term, as in KdV-type
equations, may counteract the dispersion that arises because wave
velocities vary with frequency, to give a particle equation.
An alternative approach might use "Unit velocity equations" (related
to Sum[Ax^2/A]x=0, which differentiates to Sum[A Axx-Ax^2]=0, with
time included in the summation variables), which allow non-dispersing
pulses because velocity is independent of frequency. These could have
some of the velocity in "rolled-up" (Kaluza-Klein) dimensions for
particles with mass, the remaining velocity being in space dimensions;
massless particles moving at the speed of light. I have made
negligible progress with this concept, despite experimenting with
non-linear equations (using Mathematica) for many years, looking for
stable compact travelling multi-dimensional multi-field pulses.
A broadened KdV equation can be written (in conservative form) as
[n^2(r+m) A^(1+m)]t + [ [A^(i-k) [A^k]x/k]x + n(1+nr)A^(1+2/n)Ax+(r-1)Ax^2/A]x = 0; the KP term [A^(1-j) Ay/j]y can be included.
This has all the standard Sech^n solitonic solutions, and their
repeating lattice JacobiCN^n solutions when r=1, m=0, n=1 or 2.
Rescaling is needed when n=-1 or -2. This gives an equation solved
by a rational pulse equation
A = (a/(1+ (a*(kx*(x+\xi-kx*t)^2+ky*(y+yi-ky*t)^2))/2))^n
when it is rescaled to
(1+n)At A^n - Axxx - Ayyy +(1+1/n)(2(Axx Ax+Ayy Ay)-(Ax^3+Ay^3)/A)/A=0,
and a modulated Gaussian pulse
A=E^(ao-(jx*(ax+t*jx^2-x))^2-(jy*(ay+t*jy^2-y))^2+i*(-(jx^4+jy^4)*t +jx^2*x+jy^2*y))
when it is rescaled to
4At-Axxx-Ayyy+(Ax Axx+AyAyy)/A=0.
There is a repeating lattice Sin[kx-kx^3 t]^p solution which becomes a
gaussian pulse when p->infinity, analogous to the JacobiCN pulses
becoming Sech pulses.
In each case, any number of space-like directions X={x,y,z..}can be
included. There may be stable solitons for very specific combinations
of parameters, but I have NOT been able to find them. When I first
discovered these different pulse shapes, I conjectured that rational
pulses (with square-law decay) might be the massive hadrons, gaussian
pulses the massless photons & neutrinos etc., and sech pulses the
point-like low-mass electrons etc..
When i="ii"th root of unity (not just the fourth root), with ii>2,
the modulated Gaussian equations have all factors (>2) of "ii" as
possible phasalities. This introduces field directions in addition to
the txyz dimensions. "ii"=12 allows 3, 6, & 12 phases as "Quark-wave"
hadrons, and 4-phase leptons.
Some of this is demonstrated in section 5 of HoopGlossary, in
http://Library.wolfram.com/infocenter/Mathsource/4894 (with more to
come in the next revision).
My hope is that a young iconoclast will take the concepts farther than
I can. As you rightly state, "it just hasn't got anywhere".
Roger Beresford.
"No-one can resist the invasion of ideas." (Victor Hugo.)
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nRoger Beresford <mail@beresford22.freeserve.co.uk> writes\n\n> Egan implies that solitonic wave packets have Heisenberg uncertainty\n>and Planck size.\n\nOh. I am imagining something different. Forgetting mass 0 particles,\nwhich are a special case, I am imagining the solitonic wave packet AS\nthe particle.\n\nSomething like the small amplitude breather here\nhttp://homepages.tversu.ru/~s000154/collision/sge_sol/sge_sol.html\nbut in more dimensions.\n\n>He does not mention the other features that make\n>solitons so appealing as particle paradigms - that they can carry\n>modulation (deBroglie waves) and that they pass each other with\n>displacement of their trajectories (mutual attraction or repulsion).\n\nOh, the latter is nice to know. None of the (very) pretty pictures have\nshown that, and its quite an important characteristic.\n\n>Attraction takes place (for KdV solitons) in an intermediate region\n>that is lower and broader than either of the well-separated packets.\n\nI\'m not sure I understand precisely what you mean here.\n\n>There are no recognizable "virtual messenger particles" to cause this\n>attraction - its all in the interaction between the terms of the\n>equation.\n\nWouldn\'t it be true to say that a solitonlike model of an elementary\nparticle would be \'analytic\'. So it would be each particle feeling (or\nbeing disturbed by) the field that is part of the other?\n\nAlso, from my gleanings from the tables of my betters, I get the\nimpression that \'virtual particles\' don\'t really exist but are the\nrepresentation of terms in an expansion. The \'full\' theory being a field\ntheory, which could fit solitons very nicely.\n\n>Missing features (for simple solitons) are combination and\n>spontaneous decay, changing the nature of the particles.\n\nI\'m somewhat surprised by this, although one might demand multi-\ndimensional models to show it. Bearing in mind the number of truly\nstable free particles is *very* limited (electron, neutrino and\nphoton?), the number of stable solutions required is rather limited.\nThat would imply that most particles are unstable solitons, and one\nwould be looking for quasi-stable solutions for these.\n\nOne imagines that quarks would only be stable in an environment with\ngreater or lesser degrees of freedom corresponding to the environment\nwithin a nucleus.\n\n>On the basis\n>that phenomena are governed by equations, there ought to be a\n>sophisticated non-linear equation that has such features.\n\nQuite, the bu**er is finding it. One might imagine that the neutrino\nwould be the simplest to model, since it seems to have spin, mass and\nlittle else.\n\n> Solitonic PDE\'s are expressible in "conservative" forms, equating\n>the sums of differentiated (conserved) terms to zero.\n\nI will believe you.\n\n>I use the common\n>convention that Ax & [F(A)]x means differentiation by "x", so n^2 At\n>+Axxx +(1+n)(2+n) A^(2/n) Ax =0 (Zabulsky 1963 with n replaced by 1/n\n>) gives the KdV and MKdV equations when n=1 or 2. The KdV, MKdV, and\n>KP (Kadometsev- Petviashvili) equations include terms Axxx & Ayy,\n>where "A" is a function of "t" ,"x", & (for KP) "y". They have\n>infinite sets of conserved properties characterized by the parameters\n>j or k, because [[A^(i-k) [A^k]x/k]x]x becomes Axxx and [A^(1-j)\n>Ay/j]y becomes Ayy after differentiation.\n\n<Oz boggles...>\n\n> The linear Schroedinger equation has a conservative form, [2 i M A]t\n>+ [hbar Ax]x= 0. Remoissenet ("Waves called Solitons", p29) shows that\n>it has modulated travelling Gaussian pulse solutions, but these\n>disperse with time - it describes "information about particles", and\n>not the stable particles themselves.\n\nFor some (unaccountable) reason this doesn\'t surprise me.\nOh, I guess because when a particle is trapped, then \'information about\na particle\' and \'the particle\' is much the same thing, and that\'s where\nits most used. On the other hand, when one gets into diffraction, where\na particle can be considered \'dispersed\', that\'s what we need.\n\nHmmm, I need to ponder this more.....\n[NB Always a bad move for an Oz]\n\n[Oz, thinking on the hoof] What we see is dispersion being dependent on\nthe (typically) mass of the particle. So photons have a rather free\nhabit of dispersing, whilst massive particles do not. I imagine that any\ndescription of a particle must be frame-independent so an electron \'at\nrest\' wrt one observer should have a field over spacetime that is the\nsame as that of a boosted observer. Now we ought to be able to consider\nsome aspect of the oscillating field of this electron as an event (peaks\nin the oscillating field for example). So a boosted observer (who will\nsee different mix of spatial and temporal fields- ie a different\nmomentum- should see these fields in a different pattern. Which goes\nback to de broglie. OTOH these fields will be distorted into a \'pancake-\nlike\' pattern and beamed in the direction of motion. I guess that just\nmeans whatever equation you use, it must be lorentz invariant. Hmm. I am\njust too ignorant about all this.\n\n>A non-linear term, as in KdV-type\n>equations, may counteract the dispersion that arises because wave\n>velocities vary with frequency, to give a particle equation.\n\nGood. But we need only enough to match reality, and I\'m not at all sure\nthat in reality particles are not dispersive. What is essential though\nis that they should remain as a discrete soliton even when widely\ndispersed.\n\n> An alternative approach might use "Unit velocity equations" (related\n>to Sum[Ax^2/A]x=0, which differentiates to Sum[A Axx-Ax^2]=0, with\n>time included in the summation variables), which allow non-dispersing\n>pulses because velocity is independent of frequency. These could have\n>some of the velocity in "rolled-up" (Kaluza-Klein) dimensions for\n>particles with mass, the remaining velocity being in space dimensions;\n>massless particles moving at the speed of light.\n\nI have to say that model is the one I \'casually use\' (in my infinite\nignorance). I\'m not totally convinced that the \'rolled up\' dimensions\nare tiny, though. I\'d quite like to have time as a rotating (in some\nsense) dimension of this sort so that when we rotate the particle by a\nboost we get to see more and more of the (rotating) time dimension\nvisible in our spatial dimension. Of course this means that each\nparticle \'carries\' its own individual time (and so by inference space),\nwhich puts a whole new magnitude to the idea of \'local\'. If you squint\nhard enough this even makes entangled particles rather less strange.\n\n>I have made\n>negligible progress with this concept, despite experimenting with\n>non-linear equations (using Mathematica) for many years, looking for\n>stable compact travelling multi-dimensional multi-field pulses.\n\nI hope it was fun though.\n\n> A broadened KdV equation can be written (in conservative form) as\n>[n^2(r+m) A^(1+m)]t + [ [A^(i-k) [A^k]x/k]x + n(1+nr)A^(1+2/n)\n>Ax+(r-1)Ax^2/A]x = 0; the KP term [A^(1-j) Ay/j]y can be included.\n>This has all the standard Sech^n solitonic solutions, and their\n>repeating lattice JacobiCN^n solutions when r=1, m=0, n=1 or 2.\n\n<Oz boggles...>\n\n> Rescaling is needed when n=-1 or -2. This gives an equation solved\n>by a rational pulse equation\n>A = (a/(1+ (a*(kx*(x+xi-kx*t)^2+ky*(y+yi-ky*t)^2))/2))^n\n>when it is rescaled to\n>(1+n)At A^n - Axxx - Ayyy +(1+1/n)(2(Axx Ax+Ayy Ay)\n>-(Ax^3+Ay^3)/A)/A=0,\n>and a modulated Gaussian pulse\n>A=E^(ao-(jx*(ax+t*jx^2-x))^2-(jy*(ay+t*jy^2-y))^2+\n>i*(-(jx^4+jy^4)*t +jx^2*x+jy^2*y))\n>when it is rescaled to\n>4At-Axxx-Ayyy+(Ax Axx+AyAyy)/A=0.\n>There is a repeating lattice Sin[kx-kx^3 t]^p solution which becomes a\n>gaussian pulse when p->infinity, analogous to the JacobiCN pulses\n>becoming Sech pulses.\n>In each case, any number of space-like directions X={x,y,z..}can be\n>included. There may be stable solitons for very specific combinations\n>of parameters, but I have NOT been able to find them. When I first\n>discovered these different pulse shapes, I conjectured that rational\n>pulses (with square-law decay) might be the massive hadrons, gaussian\n>pulses the massless photons & neutrinos etc., and sech pulses the\n>point-like low-mass electrons etc..\n\nI regret this is over my head.\n\n> When i="ii"th root of unity (not just the fourth root), with ii>2,\n>the modulated Gaussian equations have all factors (>2) of "ii" as\n>possible phasalities. This introduces field directions in addition to\n>the txyz dimensions. "ii"=12 allows 3, 6, & 12 phases as "Quark-wave"\n>hadrons, and 4-phase leptons.\n> Some of this is demonstrated in section 5 of HoopGlossary, in\n>http://Library.wolfram.com/infocenter/Mathsource/4894 (with more to\n>come in the next revision).\n>My hope is that a young iconoclast will take the concepts farther than\n>I can.\n\nI\'ve been hoping for decades, good job I never held my breath.\nBut then I suspect every physicist the world over has been hoping too.\n\n>As you rightly state, "it just hasn\'t got anywhere".\n\n<sigh>\n\n--\nOz\nThis post is worth absolutely nothing and is probably fallacious.\n\nBTOPENWORLD address about to cease. DEMON address no longer in use.\n>>Use oz@farmeroz.port995.com<<\nozacoohdb@despammed.com still functions.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Roger Beresford <mail@beresford22.freeserve.co.uk> writes
> Egan implies that solitonic wave packets have Heisenberg uncertainty
>and Planck size.
Oh. I am imagining something different. Forgetting mass particles,
which are a special case, I am imagining the solitonic wave packet AS
the particle.
Something like the small amplitude breather here
http://homepages.tversu.ru/~s000154/collision/sge_sol/sge_sol.html
but in more dimensions.
>He does not mention the other features that make
>solitons so appealing as particle paradigms - that they can carry
>modulation (deBroglie waves) and that they pass each other with
>displacement of their trajectories (mutual attraction or repulsion).
Oh, the latter is nice to know. None of the (very) pretty pictures have
shown that, and its quite an important characteristic.
>Attraction takes place (for KdV solitons) in an intermediate region
>that is lower and broader than either of the well-separated packets.
I'm not sure I understand precisely what you mean here.
>There are no recognizable "virtual messenger particles" to cause this
>attraction - its all in the interaction between the terms of the
>equation.
Wouldn't it be true to say that a solitonlike model of an elementary
particle would be 'analytic'. So it would be each particle feeling (or
being disturbed by) the field that is part of the other?
Also, from my gleanings from the tables of my betters, I get the
impression that 'virtual particles' don't really exist but are the
representation of terms in an expansion. The 'full' theory being a field
theory, which could fit solitons very nicely.
>Missing features (for simple solitons) are combination and
>spontaneous decay, changing the nature of the particles.
I'm somewhat surprised by this, although one might demand multi-
dimensional models to show it. Bearing in mind the number of truly
stable free particles is *very* limited (electron, neutrino and
photon?), the number of stable solutions required is rather limited.
That would imply that most particles are unstable solitons, and one
would be looking for quasi-stable solutions for these.
One imagines that quarks would only be stable in an environment with
greater or lesser degrees of freedom corresponding to the environment
within a nucleus.
>On the basis
>that phenomena are governed by equations, there ought to be a
>sophisticated non-linear equation that has such features.
Quite, the bu**er is finding it. One might imagine that the neutrino
would be the simplest to model, since it seems to have spin, mass and
little else.
> Solitonic PDE's are expressible in "conservative" forms, equating
>the sums of differentiated (conserved) terms to zero.
I will believe you.
>I use the common
>convention that Ax & [F(A)]x means differentiation by "x", so n^2 At>+Axxx +(1+n)(2+n) A^(2/n) Ax =0 (Zabulsky 1963 with n replaced by 1/n
>) gives the KdV and MKdV equations when n=1 or 2. The KdV, MKdV, and
>KP (Kadometsev- Petviashvili) equations include terms Axxx & Ayy,
>where "A" is a function of "t" ,"x", & (for KP) "y". They have
>infinite sets of conserved properties characterized by the parameters
>j or k, because [[A^(i-k) [A^k]x/k]x]x becomes Axxx and [A^(1-j)>Ay/j]y becomes Ayy after differentiation.
<Oz boggles...>
> The linear Schroedinger equation has a conservative form, [2 i M A]t
>+ [\hbar Ax]x= . Remoissenet ("Waves called Solitons", p29) shows that
>it has modulated travelling Gaussian pulse solutions, but these
>disperse with time - it describes "information about particles", and
>not the stable particles themselves.
For some (unaccountable) reason this doesn't surprise me.
Oh, I guess because when a particle is trapped, then 'information about
a particle' and 'the particle' is much the same thing, and that's where
its most used. On the other hand, when one gets into diffraction, where
a particle can be considered 'dispersed', that's what we need.
Hmmm, I need to ponder this more.....
[NB Always a bad move for an Oz]
[Oz, thinking on the hoof] What we see is dispersion being dependent on
the (typically) mass of the particle. So photons have a rather free
habit of dispersing, whilst massive particles do not. I imagine that any
description of a particle must be frame-independent so an electron 'at
rest' wrt one observer should have a field over spacetime that is the
same as that of a boosted observer. Now we ought to be able to consider
some aspect of the oscillating field of this electron as an event (peaks
in the oscillating field for example). So a boosted observer (who will
see different mix of spatial and temporal fields- ie a different
momentum- should see these fields in a different pattern. Which goes
back to de broglie. OTOH these fields will be distorted into a 'pancake-
like' pattern and beamed in the direction of motion. I guess that just
means whatever equation you use, it must be lorentz invariant. Hmm. I am
just too ignorant about all this.
>A non-linear term, as in KdV-type
>equations, may counteract the dispersion that arises because wave
>velocities vary with frequency, to give a particle equation.
Good. But we need only enough to match reality, and I'm not at all sure
that in reality particles are not dispersive. What is essential though
is that they should remain as a discrete soliton even when widely
dispersed.
> An alternative approach might use "Unit velocity equations" (related
>to Sum[Ax^2/A]x=0, which differentiates to Sum[A Axx-Ax^2]=0, with
>time included in the summation variables), which allow non-dispersing
>pulses because velocity is independent of frequency. These could have
>some of the velocity in "rolled-up" (Kaluza-Klein) dimensions for
>particles with mass, the remaining velocity being in space dimensions;
>massless particles moving at the speed of light.
I have to say that model is the one I 'casually use' (in my infinite
ignorance). I'm not totally convinced that the 'rolled up' dimensions
are tiny, though. I'd quite like to have time as a rotating (in some
sense) dimension of this sort so that when we rotate the particle by a
boost we get to see more and more of the (rotating) time dimension
visible in our spatial dimension. Of course this means that each
particle 'carries' its own individual time (and so by inference space),
which puts a whole new magnitude to the idea of 'local'. If you squint
hard enough this even makes entangled particles rather less strange.
>I have made
>negligible progress with this concept, despite experimenting with
>non-linear equations (using Mathematica) for many years, looking for
>stable compact travelling multi-dimensional multi-field pulses.
I hope it was fun though.
> A broadened KdV equation can be written (in conservative form) as
>[n^2(r+m) A^(1+m)]t + [ [A^(i-k) [A^k]x/k]x + n(1+nr)A^(1+2/n)>Ax+(r-1)Ax^2/A]x = 0; the KP term [A^(1-j) Ay/j]y can be included.
>This has all the standard Sech^n solitonic solutions, and their
>repeating lattice JacobiCN^n solutions when r=1, m=0, n=1 or 2.
<Oz boggles...>
> Rescaling is needed when n=-1 or -2. This gives an equation solved
>by a rational pulse equation
>A = (a/(1+ (a*(kx*(x+\xi-kx*t)^2+ky*(y+yi-ky*t)^2))/2))^n
>when it is rescaled to
>(1+n)At A^n - Axxx - Ayyy +(1+1/n)(2(Axx Ax+Ayy Ay)>-(Ax^3+Ay^3)/A)/A=0,
>and a modulated Gaussian pulse
>A=E^(ao-(jx*(ax+t*jx^2-x))^2-(jy*(ay+t*jy^2-y))^2+>i*(-(jx^4+jy^4)*t +jx^2*x+jy^2*y))
>when it is rescaled to
>4At-Axxx-Ayyy+(Ax Axx+AyAyy)/A=0.
>There is a repeating lattice Sin[kx-kx^3 t]^p solution which becomes a
>gaussian pulse when p->infinity, analogous to the JacobiCN pulses
>becoming Sech pulses.
>In each case, any number of space-like directions X={x,y,z..}can be
>included. There may be stable solitons for very specific combinations
>of parameters, but I have NOT been able to find them. When I first
>discovered these different pulse shapes, I conjectured that rational
>pulses (with square-law decay) might be the massive hadrons, gaussian
>pulses the massless photons & neutrinos etc., and sech pulses the
>point-like low-mass electrons etc..
I regret this is over my head.
> When i="ii"th root of unity (not just the fourth root), with ii>2,
>the modulated Gaussian equations have all factors (>2) of "ii" as
>possible phasalities. This introduces field directions in addition to
>the txyz dimensions. "ii"=12 allows 3, 6, & 12 phases as "Quark-wave"
>hadrons, and 4-phase leptons.
> Some of this is demonstrated in section 5 of HoopGlossary, in
>http://Library.wolfram.com/infocenter/Mathsource/4894 (with more to
>come in the next revision).
>My hope is that a young iconoclast will take the concepts farther than
>I can.
I've been hoping for decades, good job I never held my breath.
But then I suspect every physicist the world over has been hoping too.
>As you rightly state, "it just hasn't got anywhere".
<sigh>
--
Oz
This post is worth absolutely nothing and is probably fallacious.
BTOPENWORLD address about to cease. DEMON address no longer in use.
>>Use oz@farmeroz.port995.com<<
ozacoohdb@despammed.com still functions.
tessel@tum.bot
Jul15-04, 03:57 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nOn Wed, 14 Jul 2004, Roger Beresford wrote:\n\n> Egan implies that solitonic wave packets have Heisenberg uncertainty and\n> Planck size.\n\nI don\'t understand this, but since Greg Egan sometimes posts here, maybe\nhe can explain.\n\n> He does not mention the other features that make solitons so appealing\n> as particle paradigms - that they can carry modulation (deBroglie waves)\n> and that they pass each other with displacement of their trajectories\n> (mutual attraction or repulsion). Attraction takes place (for KdV\n> solitons) in an intermediate region that is lower and broader than\n> either of the well-separated packets.\n\n\n> There are no recognizable "virtual messenger particles" to cause this\n> attraction - its all in the interaction between the terms of the\n> equation.\n\nYou and Oz should enjoy some of the pictures in the "picture book"\n\nauthor = {E. Atlee Jackson},\ntitle = {Perspectives of Nonlinear Dynamics},\nnote = {Two Volumes},\npublisher = {Cambridge University Press},\nyear = 1991}\n\nSee the discussion of the interaction of two dimensional solitonic ridges\nand an intriguing "resonance condition"! Here, it seems, we -can- have\nsomething that looks, if you squint real hard, just a bit like "virtual\nparticles" mediating an "interaction".\n\n"Resonance" may bring in another idea I\'m intrigued by: possible\nconnections with number theory.\n\nLet me try to add a few explanatory glosses to Roger\'s post which may help\nothers to follow along.\n\n> Missing features (for simple solitons) are combination and\n> spontaneous decay, changing the nature of the particles.\n\n> On the basis that phenomena are governed by equations, there ought to be\n> a sophisticated non-linear equation that has such features.\n\nI probably should have said that the "harmonic maps" I mentioned -have-\nbeen used to formulate possible field theories ("nonlinear sigma models"),\nand these do directly involve "soliton PDEs". The book\n\neditor = {A. P. Fordy and J. C. Wood},\ntitle = {Harmonic Maps and Integrable Systems},\nseries = {Aspects of Mathematics},\nvolume = {E23},\npublisher = {Vieweg},\nyear = 1994,\nnote = {available at\nhttp://www.amsta.leeds.ac.uk/Pure/staff/wood/FordyWood/contents.html}}\n\nhas a chapter on this stuff.\n\n> Solitonic PDE\'s are expressible in "conservative" forms, equating\n> the sums of differentiated (conserved) terms to zero.\n\nI think I already mentioned that the KdV can be written\n\nu_t = u_(xxx) + 6 u u_x = @/@x (u_xx + 3 u^2)\n\nHere, we can interpret the LHS as a "density" and the RHS as a "flux".\nIntegrating over the real line and assuming that u(x,t) is "asympotically\nand sufficiently quickly vanishing", we conclude that for each particular\nsolution, the number\n\nM(t) = int_{-infty}^{infty} u(x,t) dx\n\ndoes not change as t varies. In the shallow water wave interpretation of\nthe sech^2 solutions, this number can be regarded as the "mass" of an\nisolated soliton. It is interesting to try to interpret the higher\nconserved quantities physically.\n\nTurning this around, the higher conservation laws I mentioned can\napparently be considered similarly, once we introduce suitable "higher KdV\nequations".\n\n> KdV, MKdV, and KP (Kadometsev- Petviashvili) equations\n\nProbably most readers are getting a bit confused by all these names. To\nrecapitulate:\n\n* KP is a fourth order PDE, the basic two dimensional soliton equation\ndescribing shallow water waves, among other things.\n\n* We can always change to another coordinate chart (e.g. radial) and this\nwill usually change the appearance of the equation. If not, we have found\na point symmetry.\n\n* Boussinesq is another fourth order PDE, the basic one dimensional\nsoliton equation describing shallow water waves, the one dimensional\nversion of KP.\n\n* Recall we can "factor" the ordinary wave operator\n\n-D_(tt) + D_(xx) = -(-D_t + D_x)(D_t + D_x)\n\nwhere D_t + D_x allows only unidirectional propagation. Analogously, if\nwe transform to a chart moving with the principal peak of some\nmultisoliton solution to Boussinesq, we obtain right or left moving\nversions of the KdV.\n\n* The FPU lattice is a discrete model which can be reduced, in a kind of\ncontinuum limit, to Boussinesq and then to KdV.\n\n* Miura noticed that there is a Baecklund morphism mapping solutions of\nthe MKdV\n\nv_t = v_(xxx) + v^2 v_x\n\nto solutions of the KdV\n\nu_t = u_(xxx) + u u_x\n\nThese turn out to form a kind of unified pair, and their tight\ninterrelationship motivates the Lax formulation, which is the key to all\nknown unified treatments of "soliton equations".\n\n* There is also a Baecklund automorphism on the space of solutions of the\nKdV, or rather its potential form.\n\n* The KdV/MKdV (and many other PDEs) have "potential forms". The KdV\nitself is not the Euler-Lagrange equation of a Lagrangian, but its\npotential form is. From\n\nL = [U_(xx)]^2/2 - [U_x]^3/6 + U_x U_t/2\n\nwe compute\n\n0 = E[L] = D_x^2 @L/@U_(xx) - D_x @L/@U_x - D_t @L/@U_t\n\n= U_(xxxx) + U_x U_(xx) - U_(tx)\n\nIf we put u = U_x in this equation, we recover\n\nu_(xxx) + u u_x - u_t\n\n(a form of the KdV).\n\nIf we can write the EL equation in the form\n\n(U_t)_x = [U_(xxx) + U_x^2/2 ]_x\n\nwe see that this implies the simpler equation\n\nU_t = U_(xxx) + [U_x]^2/2\n\nwhich we can take as the potential form.\n\n* The KdV also has a Hamiltonian form, in fact more than one. This is\nanother hallmark of a true soliton equation.\n\n> The linear Schroedinger equation has a conservative form, [2 i M A]t\n> + [hbar Ax]x= 0. Remoissenet ("Waves called Solitons", p29) shows that\n> it has modulated travelling Gaussian pulse solutions, but these\n> disperse with time - it describes "information about particles", and\n> not the stable particles themselves.\n\nApparently the -nonlinear- Schroedinger equation has something to do with\nenvelopes of wave packets, e.g. in fiber optic cables, but I haven\'t yet\nstudied this. It seems that some papers cited in the book I cited above\ndiscuss this.\n\n> A non-linear term, as in KdV-type equations, may counteract the\n> dispersion that arises because wave velocities vary with frequency, to\n> give a particle equation.\n\nI should have said more about this "nonlinear term" and "dispersive term"\nbusiness. Compare:\n\nu_t + u u_x + delta u_(xxx) = 0 (KdV)\n^^^^^ ^^^^^^^^^^^^\nnonlinear dispersive\nadvection term\nterm\n\nu_t + u u_x + delta u_(xx) = 0 (Burger\'s)\n^^^^^^^^^^^^\ndiffusion\nterm\n\nNote that the difference between KdV and Burger\'s is that KdV has a\ndispersion term and Burgers has a diffusion term. To understand this,\ncompare two limiting cases:\n\nu_t + delta u_(xxx) = 0 (linearized KdV)\n\nu_t + delta u_(xx) = 0 (linearized Burger\'s, or heat equation)\n\nThe heat equation (aka Fick\'s equation) is the classic diffusion equation.\nThe fundamental solution is a Gaussian which spreads out (diffuses).\n\nThe linearized KdV is a classic wave equation illustrating dispersion.\nThat is, traveling wave solutions of form\n\nu(x,t) = f(kx - omega t)\n\nare only possible if the dispersion relation\n\nomega = k-k^3\n\nis satisfied. Here, k is wavenumber and omega is frequency (these are\ncompeting "spatial" and "temporal" notions of "oscillation rate"). The\ntwo characteristic speeds are\n\nphase velocity = omega/k = 1-k^2\n\ngroup velocity = domega/dk = 1-3k^2\n\n(Note the group velocity is smaller than the phase velocity.) Recall that\ndispersion means that different frequency components move at different\nspeeds; thus, wavecrests tend to "disperse".\n\nThe traveling wave solution of the linearized KdV has the form of an Airy\nfunction: a main wavecrest followed by a rapidly diminishing wavetrain.\nIn the full KdV, this explains the FPU behavior I described in my last\npost: start with a sine wave; this wavecrest steepens, then just as it is\nabout to break, the dispersive term intervenes and creates a "dispersing\nwavetrain of solitons", which looks (not by coincidence) much like an Airy\nfunction.\n\nNow you know something about what makes the KdV and Burger\'s equation\n-different-. What about the bit they have in common? Well, that is the\n"nonlinear advection equation"\n\nu_t + u u_x = 0\n\nI keep saying that this admits traveling waves only for a finite length of\ntime, because any initial wave profile will evolve as the wave moves so\nthat the crest becomes steeper until after a finite time, the solution\nbreaks down (tries to become multiple valued). This is very simple to\nunderstand: plug in the Ansatz f(x-u t):\n\nf\'(x - u t) (-t u_t - u) + u f\'(x-u t) (1-t u_x)\n\n= f\'(x- u t) [ u - u - t (u u_x + u_t) ]\n\nso this Ansatz defines a solution implicitly. But graphically it says\nthis: start with any wave profile, and as t increases, to obtain the\nevolution of the wave profile, just move points on the graph\n-horizontally- to the right with speed proportional to their height. Of\ncourse, no matter what profile we started with, as long as it was nonzero,\nin a finite time our wave must "break".\n\nThus, we say that in the KdV we balance two opposing effects: nonlinear\nadvection, which tends to make wavecrests steepen and break, and\ndispersion, which tends to make wavecrests "disperse" into a train of\ndistinct frequency components traveling at different speeds. In the\n"1-soliton solution" (Scott Russell\'s sech^2 thingie), we have a perfect\nbalance and the soliton propogates without changing shape.\n\nFurther insight comes when we compute the point symmetries (and more\nexotic symmetries) of all these equations and use Lie\'s methods to study\ntheir solutions.\n\nSo, we might say that KdV is a nonlinear dispersion-advection equation,\nwhile Burger\'s is a nonlinear diffusion-advection equation.\n\n> This has all the standard Sech^n solitonic solutions, and their\n> repeating lattice JacobiCN^n solutions when r=1, m=0, n=1 or 2.\n\nE.g. in the KdV, we have a variety of traveling wave solutions, by no\nmeans all of which represent solitons. In particular, we have the\n"cnoidal waves" found by Korteweg and de Vries themselves. As I said,\ntheir existence is plausible from a phase plane analysis, but to derive\nthem you need more knowledge of Jacobi elliptic functions than is common\nat the beginning of the 21st century! As the period of these periodic\nsolutions increases without limit, we obtain the 1-soliton, the sech^2\nsolution found empirically by Scott Russell. This corresponds to the\nsepatrix in the phase portrait.\n\n> My hope is that a young iconoclast will take the concepts farther than I\n> can. As you rightly state, "it just hasn\'t got anywhere".\n\nGrrr! Only true (AFAIK) if you are trying to use solitons to say\nsomething profound about particle physics. But of course, this is not an\nunnatural goal, and may even be laudable.\n\nThe obvious question is probably: have you tried to put your generalized\nKdV equations into Lax pair form? Does the Uhlenbeck-Terng loop group\nformulation help?\n\n(Female iconoclasts take note: Uhlenbeck and Terng happen to be women, so\nyou have no excuse for not joining the hunt.)\n\n> "No-one can resist the invasion of ideas." (Victor\n\nIn a mostly serious (and highly politically charged) novel, Invasion of\nthe Sea (recently republished in the U.S.), Jules Verne pokes fun at this\naphorism of his distinguished colleague.\n\n"T. Essel" (hiding somewhere in cyberspace)\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Wed, 14 Jul 2004, Roger Beresford wrote:
> Egan implies that solitonic wave packets have Heisenberg uncertainty and
> Planck size.
I don't understand this, but since Greg Egan sometimes posts here, maybe
he can explain.
> He does not mention the other features that make solitons so appealing
> as particle paradigms - that they can carry modulation (deBroglie waves)
> and that they pass each other with displacement of their trajectories
> (mutual attraction or repulsion). Attraction takes place (for KdV
> solitons) in an intermediate region that is lower and broader than
> either of the well-separated packets.
> There are no recognizable "virtual messenger particles" to cause this
> attraction - its all in the interaction between the terms of the
> equation.
You and Oz should enjoy some of the pictures in the "picture book"
author = {E. Atlee Jackson},
title = {Perspectives of Nonlinear Dynamics},
note = {Two Volumes},
publisher = {Cambridge University Press},
year = 1991}
See the discussion of the interaction of two dimensional solitonic ridges
and an intriguing "resonance condition"! Here, it seems, we -can- have
something that looks, if you squint real hard, just a bit like "virtual
particles" mediating an "interaction".
"Resonance" may bring in another idea I'm intrigued by: possible
connections with number theory.
Let me try to add a few explanatory glosses to Roger's post which may help
others to follow along.
> Missing features (for simple solitons) are combination and
> spontaneous decay, changing the nature of the particles.
> On the basis that phenomena are governed by equations, there ought to be
> a sophisticated non-linear equation that has such features.
I probably should have said that the "harmonic maps" I mentioned -have-
been used to formulate possible field theories ("nonlinear \sigma models"),
and these do directly involve "soliton PDEs". The book
editor = {A. P. Fordy and J. C. Wood},
title = {Harmonic Maps and Integrable Systems},
series = {Aspects of Mathematics},
volume = {E23},
publisher = {Vieweg},
year = 1994,
note = {available at
http://www.amsta.leeds.ac.uk/Pure/staff/wood/FordyWood/contents.html}}
has a chapter on this stuff.
> Solitonic PDE's are expressible in "conservative" forms, equating
> the sums of differentiated (conserved) terms to zero.
I think I already mentioned that the KdV can be written
u_t = u_(xxx) + 6 u u_x = @/@x (u_{xx} + 3 u^2)
Here, we can interpret the LHS as a "density" and the RHS as a "flux".
Integrating over the real line and assuming that u(x,t) is "asympotically
and sufficiently quickly vanishing", we conclude that for each particular
solution, the number
M(t) = \int_{-\infty}^{\infty} u(x,t) dx
does not change as t varies. In the shallow water wave interpretation of
the sech^2 solutions, this number can be regarded as the "mass" of an
isolated soliton. It is interesting to try to interpret the higher
conserved quantities physically.
Turning this around, the higher conservation laws I mentioned can
apparently be considered similarly, once we introduce suitable "higher KdV
equations".
> KdV, MKdV, and KP (Kadometsev- Petviashvili) equations
Probably most readers are getting a bit confused by all these names. To
recapitulate:
* KP[/itex] is a fourth order PDE, the basic two dimensional soliton equation
describing shallow water waves, among other things.
* We can always change to another coordinate chart (e.g. radial) and this
will usually change the appearance of the equation. If not, we have found
a point symmetry.
* Boussinesq is another fourth order PDE, the basic one dimensional
soliton equation describing shallow water waves, the one dimensional
version of KP.
* Recall we can "factor" the ordinary wave operator
-D_(tt) + D_(xx) = -(-D_t + D_x)(D_t + D_x)
where D_t + D_x allows only unidirectional propagation. Analogously, if
we transform to a chart moving with the principal peak of some
multisoliton solution to Boussinesq, we obtain right or left moving
versions of the KdV.
* The FPU lattice is a discrete model which can be reduced, in a kind of
continuum limit, to Boussinesq and then to KdV.
* Miura noticed that there is a Baecklund morphism mapping solutions of
the MKdV
v_t = v_(xxx) + v^2 v_x
to solutions of the KdV
u_t = u_(xxx) + u u_x
These turn out to form a kind of unified pair, and their tight
interrelationship motivates the Lax formulation, which is the key to all
known unified treatments of "soliton equations".
* There is also a Baecklund automorphism on the space of solutions of the
KdV, or rather its potential form.
* The KdV/MKdV (and many other PDEs) have "potential forms". The KdV
itself is not the Euler-Lagrange equation of a Lagrangian, but its
potential form is. From
L = [U_(xx)]^2/2 - [U_x]^3/6 + U_x U_t/2
we compute
= E[L] = D_x^2 @L/@U_(xx) - D_x @L/@U_x - D_t @L/@U_t= U_(xxxx) + U_x U_(xx) - U_(tx)
If we put u = U_x in this equation, we recover
u_(xxx) + u u_x - u_t
(a form of the KdV).
If we can write the EL equation in the form
(U_t)_x = [U_(xxx) + U_x^2/2 ]_x
we see that this implies the simpler equation
U_t = U_(xxx) + [U_x]^2/2
which we can take as the potential form.
* The KdV also has a Hamiltonian form, in fact more than one. This is
another hallmark of a true soliton equation.
> The linear Schroedinger equation has a conservative form, [2 i M A]t
> + [\hbar Ax]x= . Remoissenet ("Waves called Solitons", p29) shows that
> it has modulated travelling Gaussian pulse solutions, but these
> disperse with time - it describes "information about particles", and
> not the stable particles themselves.
Apparently the -nonlinear- Schroedinger equation has something to do with
envelopes of wave packets, e.g. in fiber optic cables, but I haven't yet
studied this. It seems that some papers cited in the book I cited above
discuss this.
> A non-linear term, as in KdV-type equations, may counteract the
> dispersion that arises because wave velocities vary with frequency, to
> give a particle equation.
I should have said more about this "nonlinear term" and "dispersive term"
business. Compare:
u_t + u u_x + \delta u_(xxx) = (KdV)
^^^^^ ^^^^^^^^^^^^
nonlinear dispersive
advection term
term
u_t + u u_x + \delta u_(xx) = (Burger's)
^^^^^^^^^^^^
diffusion
term
Note that the difference between KdV and Burger's is that KdV has a
dispersion term and Burgers has a diffusion term. To understand this,
compare two limiting cases:
u_t + \delta u_(xxx) = (linearized KdV)
u_t + \delta u_(xx) = (linearized Burger's, or heat equation)
The heat equation (aka Fick's equation) is the classic diffusion equation.
The fundamental solution is a Gaussian which spreads out (diffuses).
The linearized KdV is a classic wave equation illustrating dispersion.
That is, traveling wave solutions of form
u(x,t) = f(kx - \omega t)
are only possible if the dispersion relation
\omega = k-k^3
is satisfied. Here, k is wavenumber and \omega is frequency (these are
competing "spatial" and "temporal" notions of "oscillation rate"). The
two characteristic speeds are
phase velocity = \omega/k = 1-k^2
group velocity = domega/dk = 1-3k^2
(Note the group velocity is smaller than the phase velocity.) Recall that
dispersion means that different frequency components move at different
speeds; thus, wavecrests tend to "disperse".
The traveling wave solution of the linearized KdV has the form of an Airy
function: a main wavecrest followed by a rapidly diminishing wavetrain.
In the full KdV, this explains the FPU behavior I described in my last
post: start with a sine wave; this wavecrest steepens, then just as it is
about to break, the dispersive term intervenes and creates a "dispersing
wavetrain of solitons", which looks (not by coincidence) much like an Airy
function.
Now you know something about what makes the KdV and Burger's equation
-different-. What about the bit they have in common? Well, that is the
"nonlinear advection equation"
u_t + u u_x =
I keep saying that this admits traveling waves only for a finite length of
time, because any initial wave profile will evolve as the wave moves so
that the crest becomes steeper until after a finite time, the solution
breaks down (tries to become multiple valued). This is very simple to
understand: plug in the Ansatz f(x-u t):f'(x - u t) (-t u_t - u) + u f'(x-u t) (1-t u_x)= f'(x- u t) [ u - u - t (u u_x + u_t) ]
so this Ansatz defines a solution implicitly. But graphically it says
this: start with any wave profile, and as t increases, to obtain the
evolution of the wave profile, just move points on the graph
-horizontally- to the right with speed proportional to their height. Of
course, no matter what profile we started with, as long as it was nonzero,
in a finite time our wave must "break".
Thus, we say that in the KdV we balance two opposing effects: nonlinear
advection, which tends to make wavecrests steepen and break, and
dispersion, which tends to make wavecrests "disperse" into a train of
distinct frequency components traveling at different speeds. In the
"1-soliton solution" (Scott Russell's sech^2 thingie), we have a perfect
balance and the soliton propogates without changing shape.
Further insight comes when we compute the point symmetries (and more
exotic symmetries) of all these equations and use Lie's methods to study
their solutions.
So, we might say that KdV is a nonlinear dispersion-advection equation,
while Burger's is a nonlinear diffusion-advection equation.
> This has all the standard [itex]Sech^n solitonic solutions, and their
> repeating lattice JacobiCN^n solutions when r=1, m=0, n=1 or 2.
E.g. in the KdV, we have a variety of traveling wave solutions, by no
means all of which represent solitons. In particular, we have the
"cnoidal waves" found by Korteweg and de Vries themselves. As I said,
their existence is plausible from a phase plane analysis, but to derive
them you need more knowledge of Jacobi elliptic functions than is common
at the beginning of the 21st century! As the period of these periodic
solutions increases without limit, we obtain the 1-soliton, the sech^2
solution found empirically by Scott Russell. This corresponds to the
sepatrix in the phase portrait.
> My hope is that a young iconoclast will take the concepts farther than I
> can. As you rightly state, "it just hasn't got anywhere".
Grrr! Only true (AFAIK) if you are trying to use solitons to say
something profound about particle physics. But of course, this is not an
unnatural goal, and may even be laudable.
The obvious question is probably: have you tried to put your generalized
KdV equations into Lax pair form? Does the Uhlenbeck-Terng loop group
formulation help?
(Female iconoclasts take note: Uhlenbeck and Terng happen to be women, so
you have no excuse for not joining the hunt.)
> "No-one can resist the invasion of ideas." (Victor
In a mostly serious (and highly politically charged) novel, Invasion of
the Sea (recently republished in the U.S.), Jules Verne pokes fun at this
aphorism of his distinguished colleague.
"T. Essel" (hiding somewhere in cyberspace)
tessel@tum.bot
Jul15-04, 03:57 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Tue, 13 Jul 2004, Robert Shaw wrote:\n\n> > Coming back to one of our questions: Are there many "soliton\n> > equations", or are the KdV and friends somehow incredibly special?\n>\n> Is it possible to start from a suitable infinite group and deduce a set\n> of PDEs with that \'hidden\' automorphism group, presumably with\n> soliton behaviour.\n\nThis is a very natural question, but right now I don\'t know the answer.\nI am myself still studying the sources I cited, but I don\'t think any of\nthem -directly- address this question. However, I sense that this\nquestion, or similar ones, are "in the background" of these papers.\n\nPerhaps once I\'ve more fully absorbed what I\'ve been reading, and have\ngained more experience with the relevant computations, I\'ll be able to say\nmore.\n\n> I\'d assume all such PDEs would be equivalent in some useful sense, maybe\n> under Backland transforms.\n\nYou may well be correct. To realize the full benefit of the papers I\ncited, one clearly needs to work through many computations, e.g. seeing in\ndetail how the Lax formulation works out not only for the KdV but for e.g.\nthe sine-Gordon equation. Once I\'ve done enough of these computations, I\nmay have a better sense of a relevant comment I\'ve seen, to the effect\nthat while initially it was not clear whether "soliton equations" are\ncommon or limited to just a few core examples (like the KdV, sine-Gordon,\nand NLS), a concensus seems to be emerging that the answer is "both".\nThat is, there are many different PDEs which turn out to exhibit all the\nhallmarks of a soliton equation which I listed (including a Baecklund\nautomorphism on the solution space), but the core examples seem to be in\nsome sense "universal".\n\n> If we started from a different group to the KdV\'s, would we get a\n> different set of PDEs, or can only that one group be the \'hidden\'\n> infinite-dimensional symmetry group of a PDE?\n\nA very good question. Your expectation that understanding the symmetry\ngroup should hold the key to ultimate understanding of any mystery is\nfully in keeping with the point of view advocated in the review paper by\nPalais, so you might want to email him to see he suggests. (I\'d be very\ninterested to learn what you find out!)\n\n"T. Essel" (hiding somewhere in cyberspace)\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Tue, 13 Jul 2004, Robert Shaw wrote:
> > Coming back to one of our questions: Are there many "soliton
> > equations", or are the KdV and friends somehow incredibly special?
>
> Is it possible to start from a suitable infinite group and deduce a set
> of PDEs with that 'hidden' automorphism group, presumably with
> soliton behaviour.
This is a very natural question, but right now I don't know the answer.
I am myself still studying the sources I cited, but I don't think any of
them -directly- address this question. However, I sense that this
question, or similar ones, are "in the background" of these papers.
Perhaps once I've more fully absorbed what I've been reading, and have
gained more experience with the relevant computations, I'll be able to say
more.
> I'd assume all such PDEs would be equivalent in some useful sense, maybe
> under Backland transforms.
You may well be correct. To realize the full benefit of the papers I
cited, one clearly needs to work through many computations, e.g. seeing in
detail how the Lax formulation works out not only for the KdV but for e.g.
the sine-Gordon equation. Once I've done enough of these computations, I
may have a better sense of a relevant comment I've seen, to the effect
that while initially it was not clear whether "soliton equations" are
common or limited to just a few core examples (like the KdV, sine-Gordon,
and NLS), a concensus seems to be emerging that the answer is "both".
That is, there are many different PDEs which turn out to exhibit all the
hallmarks of a soliton equation which I listed (including a Baecklund
automorphism on the solution space), but the core examples seem to be in
some sense "universal".
> If we started from a different group to the KdV's, would we get a
> different set of PDEs, or can only that one group be the 'hidden'
> infinite-dimensional symmetry group of a PDE?
A very good question. Your expectation that understanding the symmetry
group should hold the key to ultimate understanding of any mystery is
fully in keeping with the point of view advocated in the review paper by
Palais, so you might want to email him to see he suggests. (I'd be very
interested to learn what you find out!)
"T. Essel" (hiding somewhere in cyberspace)
tessel@tum.bot
Jul15-04, 03:57 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Tue, 13 Jul 2004, Oz wrote, regarding the webpage\n\nhttp://homepages.tversu.ru/~s000154/collision/main.html\n\n> Are just wonderful. I particularly like the effect of amplitude, and the\n> \'low amplitude\' waves, which do look awfully like a wavebundle, must\n> surely be the sort of thing a particle should be like.\n\nWhich animation on that site are you looking at?\n\n> Obviously work on solitons hasn\'t stopped, it just hasn\'t got anywhere.\n\nWhoa! Stop press! You previously said something else which made me think\nthat you somehow regard soliton theory as some kind of "failure" because\nit hasn\'t yet lead to a universally accepted unified field theory. I\npreviously tried to correct this serious misconception, but evidently I\nfailed on my first attempt. Now I feel like screaming:\n\nTHE DEVELOPMENT OF SOLITON THEORY IS UNIVERSALLY REGARDED AS ONE OF THE\nMOST IMPORTANT DEVELOPMENTS IN PHYSICS/APPLIED MATH IN THE PAST 50\nYEARS! IN SCIENCE, SUCH APPROBATION IS THE EXACT OPPOSITE OF FAILURE!\n\nOK, that feels better :-)\n\nBut let me say it again, more sedately. Soliton theory stands as one of\nthe great achievements (even -the- great achievement) of "nonlinear\nscience" to date. This assessment is in no way diminished by the fact\nthat the theory is perhaps not yet "fully complete" or that deep mysteries\nlike the Painleve conjecture remain unsolved. It is apparently true that\nthe very real and very important successes of soliton theory do not, as\nyet, include universally recognized deep connections with -particle\nphysics-. But, since there is so much more in heaven and Earth than\nsubatomic particles, it would be extremely silly to be so disappointed by\nthis situation (which may be only temporary) as to declare soliton theory\na "failure".\n\nI may get a chance to try to say more about what you\'re looking at one\nthose web pages in a few days; we\'ll see.\n\n"T. Essel" (hiding somewhere in cyberspace)\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Tue, 13 Jul 2004, Oz wrote, regarding the webpage
http://homepages.tversu.ru/~s000154/collision/main.html
> Are just wonderful. I particularly like the effect of amplitude, and the
> 'low amplitude' waves, which do look awfully like a wavebundle, must
> surely be the sort of thing a particle should be like.
Which animation on that site are you looking at?
> Obviously work on solitons hasn't stopped, it just hasn't got anywhere.
Whoa! Stop press! You previously said something else which made me think
that you somehow regard soliton theory as some kind of "failure" because
it hasn't yet lead to a universally accepted unified field theory. I
previously tried to correct this serious misconception, but evidently I
failed on my first attempt. Now I feel like screaming:
THE DEVELOPMENT OF SOLITON THEORY IS UNIVERSALLY REGARDED AS ONE OF THE
MOST IMPORTANT DEVELOPMENTS IN PHYSICS/APPLIED MATH IN THE PAST 50
YEARS! IN SCIENCE, SUCH APPROBATION IS THE EXACT OPPOSITE OF FAILURE!
OK, that feels better :-)
But let me say it again, more sedately. Soliton theory stands as one of
the great achievements (even -the- great achievement) of "nonlinear
science" to date. This assessment is in no way diminished by the fact
that the theory is perhaps not yet "fully complete" or that deep mysteries
like the Painleve conjecture remain unsolved. It is apparently true that
the very real and very important successes of soliton theory do not, as
yet, include universally recognized deep connections with -particle
physics-. But, since there is so much more in heaven and Earth than
subatomic particles, it would be extremely silly to be so disappointed by
this situation (which may be only temporary) as to declare soliton theory
a "failure".
I may get a chance to try to say more about what you're looking at one
those web pages in a few days; we'll see.
"T. Essel" (hiding somewhere in cyberspace)
tessel@tum.bot
Jul16-04, 08:19 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Thu, 15 Jul 2004, Oz wrote:\n\n> Forgetting mass 0 particles, which are a special case, I am imagining\n> the solitonic wave packet AS the particle.\n\nYeah, but that can\'t be right, because (among other reasons) as Roger\nsaid:\n\n> >Missing features (for simple solitons) are combination and\n> >spontaneous decay, changing the nature of the particles.\n>\n> I\'m somewhat surprised by this,\n\nYou shouldn\'t be. Recall that solitons are extremely persistent--- they\ndon\'t disperse or diffuse way, unlike single wavecrest solutions of\nordinary old wave equations. In particular, we don\'t see anything\nresembling spontaneous decay of solitons! (Or two solitons somehow\ncombining into one, like two mutually approaching billiard balls in one\ndimension in the center of mass frame.) Also, when we smash particles in\na collider, the collison typically produces a spray of new particles plus\nmassless radiation. But, as we saw, when KdV or SG solitons collide,\nafter collision the solitons emerge unscathed except for a phase change,\nso nothing like a -spray- of new particles, and there is no "radiation" in\nsight. This remains true for more than one spatial dimension.\n\nNote: I am discussing "true solitons". I think Roger may also have in\nmind certain "fake solitons" which -can- behave a bit more like colliding\nparticles. Some of these are discussed in the first chapter of one book I\ncited\n\neditor = {Gu Chaohao},\ntitle = {Soliton Theory and Its Applications},\npublisher = {Springer-Verlag},\nyear = 1990}\n\n> >He does not mention the other features that make solitons so appealing\n> >as particle paradigms - that they can carry modulation (deBroglie\n> >waves) and that they pass each other with displacement of their\n> >trajectories (mutual attraction or repulsion).\n\nYou and Roger should definitely see the pictures in the second volume of\nthe "picture book"\n\nauthor = {E. Atlee Jackson},\ntitle = {Perspectives of Nonlinear Dynamics},\nnote = {Two Volumes},\npublisher = {Cambridge University Press},\nyear = 1991}\n\nIt seems that at least one two-dimensional soliton equation apparently\n-can- admit collisions which look a bit like Feynman diagrams complete\nwith virtual particles:\n\ns s\'\n\\ /\n\\/\n||\n/\\\n/ \\\ns\' s\n\nHere, s,s\' are "solitonic ridges" and there is nothing like Lorentz\ninvariance in sight, however, so again it seems to me that this naive\ninterpretation can\'t possibly be right.\n\nAnother big problem here is that the angle matters very much. Actually,\nif you forget the naive attempt to interpret these as Feynman diagrams,\nthis is the interesting "resonance" phenomenon I mentioned.\n\nI would guess that any connection between soliton theory and particle\nphysics must involve something more subtle than the kind of naive\ninterpretation here. For example, since solitons are so stable, rather\nthan trying to identify solitons with particles themselves, I guess you\nshould try to identify them with something in particle collision scenarios\nwhich in some sense -propagates- but which is also in some sense -strictly\nconserved-. The particles themselves clearly do not fit this bill.\nThat\'s my first guess, anyway.\n\nIt is important not to approach new ideas with preconceived ideas!\nHistory shows that progress comes by paying careful attention to what the\nequations are actually saying, not what (on the basis of previous\nexperience, which is often terribly misleading in a very new situation) we\nthink we "want" them to say.\n\n> Oh, the latter is nice to know. None of the (very) pretty pictures have\n> shown that, and its quite an important characteristic.\n\nYes, they do! If I understand Roger correctly, the phase displacements\nupon collision can be regarded as the after affect of an "interaction".\nYou are probably confused by the fact that the pictures I pointed you at\nare -one dimensional-. But take a shallow pan of water and try to create\nsolitonic ridges (1-soliton solutions of KP equation). You should see\ncollisions which look like\n\n\\ /\n\\/\n|| <- nonlinear interaction zone\n/\\\n/ \\\n\n[snip rest]\n\n"T. Essel" (hiding somewhere in cyberspace)\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Thu, 15 Jul 2004, Oz wrote:
> Forgetting mass particles, which are a special case, I am imagining
> the solitonic wave packet AS the particle.
Yeah, but that can't be right, because (among other reasons) as Roger
said:
> >Missing features (for simple solitons) are combination and
> >spontaneous decay, changing the nature of the particles.
>
> I'm somewhat surprised by this,
You shouldn't be. Recall that solitons are extremely persistent--- they
don't disperse or diffuse way, unlike single wavecrest solutions of
ordinary old wave equations. In particular, we don't see anything
resembling spontaneous decay of solitons! (Or two solitons somehow
combining into one, like two mutually approaching billiard balls in one
dimension in the center of mass frame.) Also, when we smash particles in
a collider, the collison typically produces a spray of new particles plus
massless radiation. But, as we saw, when KdV or SG solitons collide,
after collision the solitons emerge unscathed except for a phase change,
so nothing like a -spray- of new particles, and there is no "radiation" in
sight. This remains true for more than one spatial dimension.
Note: I am discussing "true solitons". I think Roger may also have in
mind certain "fake solitons" which -can- behave a bit more like colliding
particles. Some of these are discussed in the first chapter of one book I
cited
editor = {Gu Chaohao},
title = {Soliton Theory and Its Applications},
publisher = {Springer-Verlag},
year = 1990}
> >He does not mention the other features that make solitons so appealing
> >as particle paradigms - that they can carry modulation (deBroglie
> >waves) and that they pass each other with displacement of their
> >trajectories (mutual attraction or repulsion).
You and Roger should definitely see the pictures in the second volume of
the "picture book"
author = {E. Atlee Jackson},
title = {Perspectives of Nonlinear Dynamics},
note = {Two Volumes},
publisher = {Cambridge University Press},
year = 1991}
It seems that at least one two-dimensional soliton equation apparently
-can- admit collisions which look a bit like Feynman diagrams complete
with virtual particles:
s s'
\ /
\/
||
/\
/ \
s' s
Here, s,s' are "solitonic ridges" and there is nothing like Lorentz
invariance in sight, however, so again it seems to me that this naive
interpretation can't possibly be right.
Another big problem here is that the angle matters very much. Actually,
if you forget the naive attempt to interpret these as Feynman diagrams,
this is the interesting "resonance" phenomenon I mentioned.
I would guess that any connection between soliton theory and particle
physics must involve something more subtle than the kind of naive
interpretation here. For example, since solitons are so stable, rather
than trying to identify solitons with particles themselves, I guess you
should try to identify them with something in particle collision scenarios
which in some sense -propagates- but which is also in some sense -strictly
conserved-. The particles themselves clearly do not fit this bill.
That's my first guess, anyway.
It is important not to approach new ideas with preconceived ideas!
History shows that progress comes by paying careful attention to what the
equations are actually saying, not what (on the basis of previous
experience, which is often terribly misleading in a very new situation) we
think we "want" them to say.
> Oh, the latter is nice to know. None of the (very) pretty pictures have
> shown that, and its quite an important characteristic.
Yes, they do! If I understand Roger correctly, the phase displacements
upon collision can be regarded as the after affect of an "interaction".
You are probably confused by the fact that the pictures I pointed you at
are -one dimensional-. But take a shallow pan of water and try to create
solitonic ridges (1-soliton solutions of KP equation). You should see
collisions which look like
\ /[/itex]
\/
|| <- nonlinear interaction zone
/\
[itex]/ \
[snip rest]
"T. Essel" (hiding somewhere in cyberspace)
Roger Beresford
Jul19-04, 03:09 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\n\nReplies to selected comments in Oz17 (message17) and Tes18 (tessel@tum\nmessage 18) etc. about my message 16. I apologize for a slow response;\nmy eyes can only take an hour at a time at the computer so reading,\nreplying, and proof-reading take several sessions.\nI am just returning to solitons after 10 years sorting out my ideas on\ngroups and algebras (undertaken to transcend the limitations of\ncomplex algebra). This thread is showing me a lot of new stuff, such\nas light bullets which are compact and have some particle-like\nproperties.\nTes might care to note that the original context was set by Oz as\n>work on solitons as a potential explanation for elementary particles.\nso the comments "it just hasn\'t got anywhere" should be read in that\ncontext.\n\nOz17 says I said\n>> Egan implies that solitonic wave packets have Heisenberg\nuncertainty\n>>and Planck size.\n> Oh. I am imagining something different. Forgetting mass 0 particles,\n>which are a special case, I am imagining the solitonic wave packet\n>AS the particle.\nSo am I. There may be different scales for "point particles" and\nothers, but I believe that the uncertainty arises because they are NOT\npoints.\n\n>>Attraction takes place (for KdV solitons) in an intermediate region\n>>that is lower and broader than either of the well-separated packets.\n>Something like the small amplitude breather?\nMy download fails to show this pic (though the kinky pics show\nnicely). I got into solitons via Richard E. Crandall\'s Mathematica for\nthe Sciences (Addison Wesley) which shows (Figs 6.2.3 & 4) a narrow\nSech^2 soliton of speed 4 height 8 overtaking a broad one of speed 1\nheight 2. They broaden in the interaction region, a col where the peak\nheight drops to 6. Cols occur in many interactions. Tes18 shows an\ninteraction between equal pulses; in Crandall\'s example the fast\nnarrow (heavy) pulse is displaced less than the broad slow pulse, -\nsomething like this:-\n\\\\ #\n\\\\ #\n\\\\ #\n\\\\\\ #\n\\|||/\n//|||\\\n#/ \\\\\\\n# \\\\\n# \\\\\nLooking back at (unpublished, long forgotten) work that I did in 1991,\nI found unit-velocity TX Gaussian solitons with quantized amplitudes\nthat interacted without phase-shift (?neutrinos?). I forgot them\nbecause the TXY & TXYZ cases did not work, and computations took many\nhours. I shall try these again, with a faster computer.\n\n>>He does not mention the other features that make\n>>solitons so appealing as particle paradigms - that they can carry\n>>modulation (deBroglie waves) and that they pass each other with\n>>displacement of their trajectories (mutual attraction or repulsion).\n>Oh, the latter is nice to know. None of the (very) pretty pictures\nhave\n>shown that, and its quite an important characteristic.\nAs Tes points out, the displacement is described as a "phase shift";\nit can be seen best when looking down, as in my illustration.\nI was too optimistic about deBroglie waves; I should have said that\nthe broadened KdV eq. has some modulated pulse solutions, which may or\nmay not be solitonic.\n\n>>Missing features (for simple solitons) are combination and\n>>spontaneous decay, changing the nature of the particles.\n>I\'m somewhat surprised by this, although one might demand\n>multi-dimensional models to show it. Bearing in mind the\n>number of truly stable free particles is *very* limited\n>(electron, neutrino and photon?), the number of stable\n>solutions required is rather limited. That would imply that\n>most particles are unstable solitons, and one\n>would be looking for quasi-stable solutions for these.\nYes. I am feeling my way towards stability, in terms of my "Dozal"\nalgebra, which has stable "orbits" looking (through my rose-tinted\nspactacles) like deBroglie waves. The quasi-stable particles should\nhave higher energies than the stable ones; I have suggested that a\n"chaotic trapdoor" effect might explain spontaneous decay. As for\nQuarks, I believe them to be the 3, 6, & 12-phase Dozal orbits, rather\nthan individual particles.\n\nGerard (in message 3) asked\n>One thing thing a soliton model of lets say the electron/muon/tau-on\n>would have to explain is the 3 different masses. This might be\n>something a soliton model could do: they predict fixed amplitudes.\n>But how could a soliton model explain that all 3 particles have\n>the same charge, and the same angular momentum?\nI still hope that my Dozal Algebra will define masses in terms of the\nvelocities in Kaluza-Klein dimensions, with charge and hypercharge\nbeing the (conserved) "sizes" of the algebra.\n\nBack to Oz 17:-\n>>I use the common convention\n>> that Ax & [F(A)]x means differentiation by "x "<snip>\n> <Oz boggles...>\nThis convention puts the equations into a compact, easily readable\nASCII form. n=1 gives At+ Axxx +6A^2 Ax=0; n=2 gives 4At+ Axxx+ 12A\nAx=0. The boggling bit is that differentiating an infinite number of\nfunctions such as A^(1-j) Ay^j or Ax^(1-k) Axx^k reduces them all to\nAyy or Axxx, losing the j & k. Hence if Ayy (or Axxx) is a term in a\nconservative equation, it has an infinite number of conserved\nproperties.\n>><snip> A broadened KdV equation can be written (in conservative\nform) as\n>>[n^2(r+m) A^(1+m)]t + [ [A^(i-k) [A^k]x/k]x + n(1+nr)A^(1+2/n)\n>>Ax+(r-1)Ax^2/A]x = 0; the KP term [A^(1-j) Ay/j]y can be included.\n>>This has all the standard Sech^n solitonic solutions, and their\n>>repeating lattice JacobiCN^n solutions when r=1, m=0, n=1 or 2.\n>><Oz boggles...><snip>\n>>I regret this is over my head.\nI was looking for differently shaped pulses that might correspond to\ndifferent particles - a rational pulse with square-law interaction,\nand a gaussian pulse with shorter range interaction than the Sech^2\nKdV pulse. I found that they fitted into the n=-1 and -2 positions of\nthe Zabulsky-type equation (which loses the original non-linear term\nfor these values of n) after adding a new conserved function. The\nJacobiCN functions eliminate the computation boundary problems by\n"wrapping round" at the boundary; they approximate ever closer to Sech\nfunctions as their width becomes smaller wrt the wrap-round length.\n\nTes18. said <long snip>\n>E.g. in the KdV, we have a variety of traveling wave solutions, by\n>no means all of which represent solitons.\nIs there a test for this? I gave up soliton studies when I could not\nestablish whether my solutions were stable and/or were composed of\nmultiple solitons. It would be nice if traveling waves could be\nanalysed (Fourier-style) into component solitons, as occurred in the\nclassical simulation; you imply that this cannot occur in general.\n\n>In particular, we have the "cnoidal waves" found by Korteweg and\n>de Vries themselves. As I said, their existence is plausible from a\n>phase plane analysis, but to derive them you need more knowledge\n>of Jacobi elliptic functions than is common at the beginning of the\n>21st century!\nThey have been in Mathematica since Version 1, which I used to\ngenerate many cnoidal KdV solutions. Whereas few of the functions that\nTes mentions in other messages are easily accessible.\nSomewhere Tes mentions "KdV solitons on the circle". I suspect that\nthis is misleading, as circles have curvature, whilst the trains of\nCnoid solitons are on an inherently flat tesselation of the complex\nplane. (OT. The OED definition of tessel@ed is "regularly chequered";\ntessel@ion is "repeated polygons without gaps or overlaps".)\n\n<snip>>The obvious question is probably: have you tried to put your\n>generalized KdV equations into Lax pair form?\nNo. In the 90\'s I did it my way, empirically - ignoring other concepts\nunless I could programme them easily. Nowadays the net drowns me in\ninformation and I do not know which concepts to test.\n>Does the Uhlenbeck-Terng loop group formulation help?\nDunno. Reference please.\n\n>In a mostly serious (and highly politically charged) novel, Invasion\nof the Sea (recently republished in the U.S.), Jules Verne pokes fun\nat the aphorism [No-one can resist the invasion of ideas] of his\ndistinguished colleague (Victor Hugo).\nMust read this. VH was wrong - ideas are easily ignored. Ever the\noptimist, I still hope that my ideas are irresistible!\n\nRoger Beresford.\n"Facts do not cease to exist because they are ignored." A Huxley.\n\nP.S. I invite anyone visiting the "Snibston Soliton" to a drink and a\nchat. I live about 8 miles away. E-mail or phone (0509 890511) me in\nadvance. R.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Replies to selected comments in Oz17 (message17) and Tes18 (tessel@tum
message 18) etc. about my message 16. I apologize for a slow response;
my eyes can only take an hour at a time at the computer so reading,
replying, and proof-reading take several sessions.
I am just returning to solitons after 10 years sorting out my ideas on
groups and algebras (undertaken to transcend the limitations of
complex algebra). This thread is showing me a lot of new stuff, such
as light bullets which are compact and have some particle-like
properties.
Tes might care to note that the original context was set by Oz as
>work on solitons as a potential explanation for elementary particles.
so the comments "it just hasn't got anywhere" should be read in that
context.
Oz17 says I said
>> Egan implies that solitonic wave packets have Heisenberg
uncertainty
>>and Planck size.
> Oh. I am imagining something different. Forgetting mass particles,
>which are a special case, I am imagining the solitonic wave packet
>AS the particle.
So am I. There may be different scales for "point particles" and
others, but I believe that the uncertainty arises because they are NOT
points.
>>Attraction takes place (for KdV solitons) in an intermediate region
>>that is lower and broader than either of the well-separated packets.
>Something like the small amplitude breather?
My download fails to show this pic (though the kinky pics show
nicely). I got into solitons via Richard E. Crandall's Mathematica for
the Sciences (Addison Wesley) which shows (Figs 6.2.3 & 4) a narrow
Sech^2 soliton of speed 4 height 8 overtaking a broad one of speed 1
height 2. They broaden in the interaction region, a col where the peak
height drops to 6. Cols occur in many interactions. Tes18 shows an
interaction between equal pulses; in Crandall's example the fast
narrow (heavy) pulse is displaced less than the broad slow pulse, -
something like this:-\\ #\\ #\\ #\\\ #\|||///|||\#/ \\\# \\# \\
Looking back at (unpublished, long forgotten) work that I did in 1991,
I found unit-velocity TX Gaussian solitons with quantized amplitudes
that interacted without phase-shift (?neutrinos?). I forgot them
because the TXY & TXYZ cases did not work, and computations took many
hours. I shall try these again, with a faster computer.
>>He does not mention the other features that make
>>solitons so appealing as particle paradigms - that they can carry
>>modulation (deBroglie waves) and that they pass each other with
>>displacement of their trajectories (mutual attraction or repulsion).
>Oh, the latter is nice to know. None of the (very) pretty pictures
have
>shown that, and its quite an important characteristic.
As Tes points out, the displacement is described as a "phase shift";
it can be seen best when looking down, as in my illustration.
I was too optimistic about deBroglie waves; I should have said that
the broadened KdV eq. has some modulated pulse solutions, which may or
may not be solitonic.
>>Missing features (for simple solitons) are combination and
>>spontaneous decay, changing the nature of the particles.
>I'm somewhat surprised by this, although one might demand
>multi-dimensional models to show it. Bearing in mind the
>number of truly stable free particles is *very* limited
>(electron, neutrino and photon?), the number of stable
>solutions required is rather limited. That would imply that
>most particles are unstable solitons, and one
>would be looking for quasi-stable solutions for these.
Yes. I am feeling my way towards stability, in terms of my "Dozal"
algebra, which has stable "orbits" looking (through my rose-tinted
spactacles) like deBroglie waves. The quasi-stable particles should
have higher energies than the stable ones; I have suggested that a
"chaotic trapdoor" effect might explain spontaneous decay. As for
Quarks, I believe them to be the 3, 6, & 12-phase Dozal orbits, rather
than individual particles.
Gerard (in message 3) asked
>One thing thing a soliton model of lets say the electron/muon/\tau-on
>would have to explain is the 3 different masses. This might be
>something a soliton model could do: they predict fixed amplitudes.
>But how could a soliton model explain that all 3 particles have
>the same charge, and the same angular momentum?
I still hope that my Dozal Algebra will define masses in terms of the
velocities in Kaluza-Klein dimensions, with charge and hypercharge
being the (conserved) "sizes" of the algebra.
Back to Oz 17:-
>>I use the common convention
>> that Ax & [F(A)]x means differentiation by "x "<snip>
> <Oz boggles...>
This convention puts the equations into a compact, easily readable
ASCII form. n=1 gives At+ Axxx +6A^2 Ax=0; n=2 gives 4At+ Axxx+ 12A
Ax=0. The boggling bit is that differentiating an infinite number of
functions such as A^(1-j) Ay^j or Ax^(1-k) Axx^k reduces them all to
Ayy or Axxx, losing the j & k. Hence if Ayy (or Axxx) is a term in a
conservative equation, it has an infinite number of conserved
properties.
>><snip> A broadened KdV equation can be written (in conservative
form) as
>>[n^2(r+m) A^(1+m)]t + [ [A^(i-k) [A^k]x/k]x + n(1+nr)A^(1+2/n)>>Ax+(r-1)Ax^2/A]x = 0; the KP term [A^(1-j) Ay/j]y can be included.
>>This has all the standard Sech^n solitonic solutions, and their
>>repeating lattice JacobiCN^n solutions when r=1, m=0, n=1 or 2.
>><Oz boggles...><snip>
>>I regret this is over my head.
I was looking for differently shaped pulses that might correspond to
different particles - a rational pulse with square-law interaction,
and a gaussian pulse with shorter range interaction than the Sech^2
KdV pulse. I found that they fitted into the n=-1 and -2 positions of
the Zabulsky-type equation (which loses the original non-linear term
for these values of n) after adding a new conserved function. The
JacobiCN functions eliminate the computation boundary problems by
"wrapping round" at the boundary; they approximate ever closer to Sech
functions as their width becomes smaller wrt the wrap-round length.
Tes18. said <long snip>
>E.g. in the KdV, we have a variety of traveling wave solutions, by
>no means all of which represent solitons.
Is there a test for this? I gave up soliton studies when I could not
establish whether my solutions were stable and/or were composed of
multiple solitons. It would be nice if traveling waves could be
analysed (Fourier-style) into component solitons, as occurred in the
classical simulation; you imply that this cannot occur in general.
>In particular, we have the "cnoidal waves" found by Korteweg and
>de Vries themselves. As I said, their existence is plausible from a
>phase plane analysis, but to derive them you need more knowledge
>of Jacobi elliptic functions than is common at the beginning of the
>21st century!
They have been in Mathematica since Version 1, which I used to
generate many cnoidal KdV solutions. Whereas few of the functions that
Tes mentions in other messages are easily accessible.
Somewhere Tes mentions "KdV solitons on the circle". I suspect that
this is misleading, as circles have curvature, whilst the trains of
Cnoid solitons are on an inherently flat tesselation of the complex
plane. (OT. The OED definition of tessel@ed is "regularly chequered";
tessel@ion is "repeated polygons without gaps or overlaps".)
<snip>>The obvious question is probably: have you tried to put your
>generalized KdV equations into Lax pair form?
No. In the 90's I did it my way, empirically - ignoring other concepts
unless I could programme them easily. Nowadays the net drowns me in
information and I do not know which concepts to test.
>Does the Uhlenbeck-Terng loop group formulation help?
Dunno. Reference please.
>In a mostly serious (and highly politically charged) novel, Invasion
of the Sea (recently republished in the U.S.), Jules Verne pokes fun
at the aphorism [No-one can resist the invasion of ideas] of his
distinguished colleague (Victor Hugo).
Must read this. VH was wrong - ideas are easily ignored. Ever the
optimist, I still hope that my ideas are irresistible!
Roger Beresford.
"Facts do not cease to exist because they are ignored." A Huxley.
P.S. I invite anyone visiting the "Snibston Soliton" to a drink and a
chat. I live about 8 miles away. E-mail or phone (0509 890511) me in
advance. R.
tessel@tum.bot
Jul20-04, 03:25 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Mon, 19 Jul 2004, Roger Beresford wrote:\n\n> Tes might care to note that the original context was set by Oz as\n> work on solitons as a potential explanation for elementary particles.\n> so the comments "it just hasn\'t got anywhere" should be read in that\n> context.\n\nActually, no, the original context was my reply to someone other than Oz,\nwho asked for a summary of "solitons in one sentence". I replied with a\npost humorously entitled "solitons in one post"--- point being that no\nsingle post could possibly do justice to this topic, much less one\nsentence. In this post, I tried to start an expository thread on solitons.\nUnfortunately, the References to all my posts seem to be hopelessly\nbollixed (is that the correct Brit-speak?), so you might not be able to\neasily find my original post using Google.\n\nOz replied asking me about something I said in this post, focusing his\nattention on my passing reference to "half-hearted attempts to connect\nparticle physics with solitons" and with certain cellular automata (for\nwhich some soliton equation might perhaps be a continuum limit). My most\nrecent posts in this thread have mostly amounted to attempts to persuade\nOz that soliton theory has been highly successful, even though AFAIK\nno-one has so far succeeded in making convincing connections with\nfundamental particle physics. But my own interest remains expositing\nsoliton theory in general, including well established links with\nsymmetries of PDEs, harmonic maps, and Hamiltonian systems; arguing with\nOz has simply been distracting me from my original goals.\n\nIn a recent post, I said:\n\n> >E.g. in the KdV, we have a variety of traveling wave solutions, by\n> >no means all of which represent solitons.\n\nLet me expand on this a bit. In the specific case of the KdV\n\nu_t + u u_x = u_(xxx)\n\nwe can focus attention on solutions which are rapidly vanishing as |x|->\ninfty. Most of these are -not- n-soliton solutions, or cnoidal waves\n(traveling waves, featuring periodic wavetrains expressed using the Jacobi\nelliptic function "cn"-- the 1-soliton solutions arise as limits from the\ncnoidal waves when the period -> infinity), or the P-I or P-II solutions I\ndiscussed using the "symmetry Ansatz method", or any of the other specific\nsolutions I have thus far discussed. (Here, P-I and P-II refers to the\nfirst two of six "Painleve transcendent" ODEs, whose solutions cannot be\nexpressed in terms of elementary functions; the associated KdV solutions\nhave the form u(x,t) = f(xi), where f is a function solving P-I or P-II,\nand xi is a simple function of x,t which is invariant under the\nappropriate flow; e.g.\n\nX = x @/@x + 3 t @/@t - 2 u @/@u\n\ngives rise to the P-II KdV solutions). But it turns out (alas, I don\'t\nhave a citation handy right now) that an "rapidly vanishing" solution to\nthe KdV must eventually resemble an n-soliton solution nonlinearly\nsuperimposed on a P-II solution, which means that it looks like a finite\nnumber of right traveling solitons (possibly having different\nheights/speeds) together with a dispersing wavetrain resembling a the\ngraph of an Airy function. The latter "component" is in some sense due to\nthe Airy function solutions of the linearized KdV\n\nu_t = u_(xxx)\n\nwhich arise when we assume u is very small. This is a linear PDE and can\nbe solved using standard Fourier transform techiniques; the "spectral\ntransform" is as I said a generalization of the Fourier transform.\nUnfortunately, I still haven\'t gotten around to discussing Baecklund\nmorphisms and nonlinear superposition laws (actually, I should first back\nway up and discuss elementary algebraic geometry stuff involving the\nhumble Ricatti equation).\n\nI already said that the three best known examples of PDEs admitting\nsoliton solutions are KdV, SG (sine-Gordon), and NLS (cubic nonlinear\nSchroedinger). I have been trying to work my way toward giving a\nreasonable answer to Gerhard\'s question about whether "soliton PDEs" are\ncommon, or whether there are just a handful of examples. AFAIK, the best\nshort answer is "both". That is, there are many soliton PDEs, but the\nthree canonical examples are in some sense "universal". But I haven\'t yet\nexplained this!\n\nGerhard also asked how one can recognize a "soliton PDE". (A soliton PDE\nis just a PDE which admits solitons as "special" solutions.) One possible\nanswer is that a PDE which is "bihamiltonian" (i.e., which can be written\nas a Hamiltonian system in two different but compatible ways) must admit\nan infinite hierarchy of Lie-Baecklund symmetries, and an infinite\nhiearchy of conservation laws. This doesn\'t quite give a genuine "test"\nfor "completely integrable Hamiltonian system" (since AFAIK there is no\nalgorithm for finding the requisite Hamiltonian formulations when they\nexist), but with some experience it can be a very efficient way to verify\nthat the KdV and various generalizations do in fact admit an infinite\nhierarchy of conservation laws. Again, I haven\'t yet explained this (and\nright now I don\'t know how to apply this to SG and NLS).\n\nVery briefly: in finite dimensional Hamiltonian systems, we usually follow\nHamilton by writing our system using canonical coordinates (this is\njustified by the famous theorem of Darboux concerning symplectic\ntwo-forms). But infinite dimensional Hamiltonian systems apparently can\'t\nbe written this way! Instead, we try to write them in the form\n\nu_t = D (delta HH)/(delta u),\n\nwhere D is a skew adjoint differential operator such as D_x, or rather a\nHamiltonian operator (has stricter conditions), and HH is a Hamiltonian\nfunctional, such as\n\nHH = int H dx\n\nwhere H is an "energy density". In this case, the Frechet derivative\nshould look like the Euler operator applied to H:\n\n(delta HH)/(delta U) = @H/@u - D_x @H/@u_x + (D_x)^2 @H/@u_(xx) - ....\n\nIn particular, for the KdV, one Hamiltonian formulation takes D_x for D\nand takes H to be the integral wrt x of the energy density I mentioned in\nan earlier post in this thread. Then (*) reduces precisely to the KdV\n\nu_t = -u u_x + u_(xxx)\n\nBut it turns out that we can also write the KdV in this form in a\ndifferent way! Namely\n\nu_t = (E delta JJ)/(delta u)\n\nwhere E is one of the operators which appears in the Lax formulation, and\nwhere JJ is (if memory serves) u^2/2, which we previously encountered as\none of our conserved densities. Then R = E D^(-1) gives a "recursion"\noperator generating an infinite hierarchy of\n\n(i) "higher KdV equations",\n\n(ii) Lie-Baecklund symmetries,\n\n(iii) conserved densities (for u rapidly vanishing).\n\nBut before I can explain this properly, I\'ll have to explain point\nsymmetries and their generalization to Lie-Baecklund symmetries,\nvariational symmetries and Noetherian conservation laws, linear operators\non function spaces (probably), Lax formulation of KdV and friends, etc.,\nnone of which which I have yet attempted...\n\nImpatient readers can read the last chapter of this wonderful book,\nhowever:\n\nauthor = {Peter J. Olver},\ntitle = {Applications of {L}ie Groups to Differential Equations},\nseries = {Graduate Texts in Mathematics},\nvolume = 107,\npublisher = {Springer-Verlag},\nyear = 1993}\n\nRoger asked:\n\n> Is there a test for this?\n\nIf I understand correctly, your question is: "when is a solution to the\nKdV is a soliton?" Well, one easy answer is it is a soliton solution if\nit has the form of the (known) n-soliton solutions for some natural number\nn!\n\nBut more generally we can ask, "when is a solution to a bihamiltonian\nsystem a soliton?" I am still trying to sort this out, but apparently in\nthe context of the Lax formulation of the KdV, there is a one-one\ncorrespondence between n eigenvalues (of a certain operator constructed in\na fairly algorithmic way) and the n-soliton solution. If possible I will\nexplain this eventually.\n\n> I gave up soliton studies when I could not establish whether my\n> solutions were stable and/or were composed of multiple solitons. It\n> would be nice if traveling waves could be analysed (Fourier-style) into\n> component solitons, as occurred in the classical simulation; you imply\n> that this cannot occur in general.\n\nI take it you are not studying the KdV itself, but a generalization.\nDoes it have a small amplitude limit which is a linear PDE? If so, does\nthis linearization admit a dispersion relation?\n\n> >In particular, we have the "cnoidal waves" found by Korteweg and\n> >de Vries themselves. As I said, their existence is plausible from a\n> >phase plane analysis, but to derive them you need more knowledge\n> >of Jacobi elliptic functions than is common at the beginning of the\n> >21st century!\n> They have been in Mathematica since Version 1, which I used to\n> generate many cnoidal KdV solutions.\n\nYes, in maple our guy the two argument defined function "JacobiCN".\n\n> Whereas few of the functions that Tes mentions in other messages are\n> easily accessible.\n\nYou mean P-I and P-II, the Painleve transcendents? If so, yes, AFAIK\nthese are not defined functions, athough they should be! In the mean\ntime, two standard sources are:\n\nauthor = {E. L. Ince},\ntitle = {Ordinary Differential Equations},\npublisher = {Dover},\nnote = {reprint of book originally published by Longmans, 1927},\nyear = 1927}\n\nauthor = {Harold T. Davis},\ntitle = {Introduction to nonlinear differential and integral equations},\npublisher = {Dover},\nyear = 1962}\n\n> Somewhere Tes mentions "KdV solitons on the circle". I suspect that\n> this is misleading, as circles have curvature,\n\nNo, I just mean "KdV on real line with stated periodicity enforced", which\nis more naturally regarded as "KdV on a circle". This is an "intrinsic\ncircle", with on embedding in any higher dimensional space needed. BTW,\n"KdV on a circle" is standard terminology.\n\n> whilst the trains of Cnoid solitons are on an inherently flat\n> tesselation of the complex plane.\n\nYes, except that we are using the real part only. But since cn is\nperiodic on real arguments, we can regard it as a singly periodic function\non R.\n\n> (OT. The OED definition of tessel@ed is "regularly chequered";\n> tessel@ion is "repeated polygons without gaps or overlaps".)\n\nMy pseudonym, "T. Essel", thus "tessel@tum.bot", is a juvenile joke (L.\ntesselatum, "tiling"), referring in part to "tiling theory" in symbolic\ndynamics, which, roughly speaking, treats a huge class of "digitized"\ndynamical systems. Examples include "shift spaces", and also "tiling\nspaces" such as the space of Penrose tilings. Symbolic dynamical systems\ncan span the spectrum from "chaotic" (the one dimensional "full shifts"\nare classic examples of chaotic dynamical systems) to "highly regular"\n("Sturmian shifts" are classic examples of one dimensional dynamical\nsystems defined on the circle). The latter should be related to\ncompletely integrable Hamiltonian systems. (For example, but Sturmian\nshifts and soliton PDEs exhibit one dimensional "quasiperiodicity".) My\ninterest in solitons was sparked by mysterious remarks by Arnold to the\neffect that he knows why "Arnold diffusion" in certain "almost Hamiltonian\nsystems" gives rise to "spatial patterns" bearing a striking resemblence\nto higher dimensional analogues of Penrose rhomb tilings.\n\n> <snip>>The obvious question is probably: have you tried to put your\n> >generalized KdV equations into Lax pair form?\n> No. In the 90\'s I did it my way, empirically - ignoring other concepts\n> unless I could programme them easily. Nowadays the net drowns me in\n> information and I do not know which concepts to test.\n\nI hope to at least explain the Lax formulation and give examples,\nincluding KdV and friends. There are many fascinating connections!\n\n> >Does the Uhlenbeck-Terng loop group formulation help?\n> Dunno. Reference please.\n\nThe article by Palais is probably the best place to start:\n\nauthor = {Richard S. Palais},\ntitle = {The Symmetries of Solitons},\njournal = {Bull. of the A. M. S.}\nvolume = {?},\nyear = {1997}\nnote = {dg-ga/9708004}}\n\n> P.S. I invite anyone visiting the "Snibston Soliton" to a drink and a\n> chat. I live about 8 miles away. E-mail or phone (0509 890511) me in\n> advance. R.\n\nI believe the legendary John Baez is currently in Hawkingland. On a\nprevious visit to Penroseville, he visited Oz, so if you\'re lucky, maybe\nhe\'ll drop in on you! (He knows much more than I do about everything, and\nalso is much better at explaining what he knows.)\n\n"T. Essel" (hiding somewhere in cyberspace)\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Mon, 19 Jul 2004, Roger Beresford wrote:
> Tes might care to note that the original context was set by Oz as
> work on solitons as a potential explanation for elementary particles.
> so the comments "it just hasn't got anywhere" should be read in that
> context.
Actually, no, the original context was my reply to someone other than Oz,
who asked for a summary of "solitons in one sentence". I replied with a
post humorously entitled "solitons in one post"--- point being that no
single post could possibly do justice to this topic, much less one
sentence. In this post, I tried to start an expository thread on solitons.
Unfortunately, the References to all my posts seem to be hopelessly
bollixed (is that the correct Brit-speak?), so you might not be able to
easily find my original post using Google.
Oz replied asking me about something I said in this post, focusing his
attention on my passing reference to "half-hearted attempts to connect
particle physics with solitons" and with certain cellular automata (for
which some soliton equation might perhaps be a continuum limit). My most
recent posts in this thread have mostly amounted to attempts to persuade
Oz that soliton theory has been highly successful, even though AFAIK
no-one has so far succeeded in making convincing connections with
fundamental particle physics. But my own interest remains expositing
soliton theory in general, including well established links with
symmetries of PDEs, harmonic maps, and Hamiltonian systems; arguing with
Oz has simply been distracting me from my original goals.
In a recent post, I said:
> >E.g. in the KdV, we have a variety of traveling wave solutions, by
> >no means all of which represent solitons.
Let me expand on this a bit. In the specific case of the KdV
u_t + u u_x = u_(xxx)
we can focus attention on solutions which are rapidly vanishing as |x|->\infty. Most of these are -not- n-soliton solutions, or cnoidal waves
(traveling waves, featuring periodic wavetrains expressed using the Jacobi
elliptic function "cn"-- the 1-soliton solutions arise as limits from the
cnoidal waves when the period -> infinity), or the P-I or P-II solutions I
discussed using the "symmetry Ansatz method", or any of the other specific
solutions I have thus far discussed. (Here, P-I and P-II refers to the
first two of six "Painleve transcendent" ODEs, whose solutions cannot be
expressed in terms of elementary functions; the associated KdV solutions
have the form u(x,t) = f(\xi), where f is a function solving P-I or P-II,
and \xi is a simple function of x,t which is invariant under the
appropriate flow; e.g.
X = x @/@x + 3 t @/@t - 2 u @/@u
gives rise to the P-II KdV solutions). But it turns out (alas, I don't
have a citation handy right now) that an "rapidly vanishing" solution to
the KdV must eventually resemble an n-soliton solution nonlinearly
superimposed on a P-II solution, which means that it looks like a finite
number of right traveling solitons (possibly having different
heights/speeds) together with a dispersing wavetrain resembling a the
graph of an Airy function. The latter "component" is in some sense due to
the Airy function solutions of the linearized KdV
u_t = u_(xxx)
which arise when we assume u is very small. This is a linear PDE and can
be solved using standard Fourier transform techiniques; the "spectral
transform" is as I said a generalization of the Fourier transform.
Unfortunately, I still haven't gotten around to discussing Baecklund
morphisms and nonlinear superposition laws (actually, I should first back
way up and discuss elementary algebraic geometry stuff involving the
humble Ricatti equation).
I already said that the three best known examples of PDEs admitting
soliton solutions are KdV, SG (sine-Gordon), and NLS (cubic nonlinear
Schroedinger). I have been trying to work my way toward giving a
reasonable answer to Gerhard's question about whether "soliton PDEs" are
common, or whether there are just a handful of examples. AFAIK, the best
short answer is "both". That is, there are many soliton PDEs, but the
three canonical examples are in some sense "universal". But I haven't yet
explained this!
Gerhard also asked how one can recognize a "soliton PDE". (A soliton PDE
is just a PDE which admits solitons as "special" solutions.) One possible
answer is that a PDE which is "bihamiltonian" (i.e., which can be written
as a Hamiltonian system in two different but compatible ways) must admit
an infinite hierarchy of Lie-Baecklund symmetries, and an infinite
hiearchy of conservation laws. This doesn't quite give a genuine "test"
for "completely integrable Hamiltonian system" (since AFAIK there is no
algorithm for finding the requisite Hamiltonian formulations when they
exist), but with some experience it can be a very efficient way to verify
that the KdV and various generalizations do in fact admit an infinite
hierarchy of conservation laws. Again, I haven't yet explained this (and
right now I don't know how to apply this to SG and NLS).
Very briefly: in finite dimensional Hamiltonian systems, we usually follow
Hamilton by writing our system using canonical coordinates (this is
justified by the famous theorem of Darboux concerning symplectic
two-forms). But infinite dimensional Hamiltonian systems apparently can't
be written this way! Instead, we try to write them in the form
u_t = D (\delta HH)/(\delta u),
where D is a skew adjoint differential operator such as D_x, or rather a
Hamiltonian operator (has stricter conditions), and HH is a Hamiltonian
functional, such as
HH = \int H dx
where H is an "energy density". In this case, the Frechet derivative
should look like the Euler operator applied to H:
(\delta HH)/(\delta U) = @H/@u - D_x @H/@u_x + (D_x)^2 @H/@u_(xx) - .[/itex]...
In particular, for the KdV, one Hamiltonian formulation takes D_x for D
and takes H to be the integral wrt x of the energy density I mentioned in
an earlier post in this thread. Then (*) reduces precisely to the KdV
u_t = -u u_x + u_(xxx)
But it turns out that we can also write the KdV in this form in a
different way! Namely
u_t = (E \delta JJ)/(\delta u)
where E is one of the operators which appears in the Lax formulation, and
where JJ is (if memory serves) u^2/2, which we previously encountered as
one of our conserved densities. Then R = E D^(-1) gives a "recursion"
operator generating an infinite hierarchy of
(i) "higher KdV equations",
(ii) Lie-Baecklund symmetries,
(iii) conserved densities (for u rapidly vanishing).
But before I can explain this properly, I'll have to explain point
symmetries and their generalization to Lie-Baecklund symmetries,
variational symmetries and Noetherian conservation laws, linear operators
on function spaces (probably), Lax formulation of KdV and friends, etc.,
none of which which I have yet attempted...
Impatient readers can read the last chapter of this wonderful book,
however:
author = {Peter J. Olver},
title = {Applications of {L}ie Groups to Differential Equations},
series = {Graduate Texts in Mathematics},
volume = 107,
publisher = {Springer-Verlag},
year = 1993}
Roger asked:
> Is there a test for this?
If I understand correctly, your question is: "when is a solution to the
KdV is a soliton?" Well, one easy answer is it is a soliton solution if
it has the form of the (known) n-soliton solutions for some natural number
n!
But more generally we can ask, "when is a solution to a bihamiltonian
system a soliton?" I am still trying to sort this out, but apparently in
the context of the Lax formulation of the KdV, there is a one-one
correspondence between n eigenvalues (of a certain operator constructed in
a fairly algorithmic way) and the n-soliton solution. If possible I will
explain this eventually.
> I gave up soliton studies when I could not establish whether my
> solutions were stable [itex]and/or were composed of multiple solitons. It
> would be nice if traveling waves could be analysed (Fourier-style) into
> component solitons, as occurred in the classical simulation; you imply
> that this cannot occur in general.
I take it you are not studying the KdV itself, but a generalization.
Does it have a small amplitude limit which is a linear PDE? If so, does
this linearization admit a dispersion relation?
> >In particular, we have the "cnoidal waves" found by Korteweg and
> >de Vries themselves. As I said, their existence is plausible from a
> >phase plane analysis, but to derive them you need more knowledge
> >of Jacobi elliptic functions than is common at the beginning of the
> >21st century!
> They have been in Mathematica since Version 1, which I used to
> generate many cnoidal KdV solutions.
Yes, in maple our guy the two argument defined function "JacobiCN".
> Whereas few of the functions that Tes mentions in other messages are
> easily accessible.
You mean P-I and P-II, the Painleve transcendents? If so, yes, AFAIK
these are not defined functions, athough they should be! In the mean
time, two standard sources are:
author = {E. L. Ince},
title = {Ordinary Differential Equations},
publisher = {Dover},
note = {reprint of book originally published by Longmans, 1927},
year = 1927}
author = {Harold T. Davis},
title = {Introduction to nonlinear differential and integral equations},
publisher = {Dover},
year = 1962}
> Somewhere Tes mentions "KdV solitons on the circle". I suspect that
> this is misleading, as circles have curvature,
No, I just mean "KdV on real line with stated periodicity enforced", which
is more naturally regarded as "KdV on a circle". This is an "intrinsic
circle", with on embedding in any higher dimensional space needed. BTW,
"KdV on a circle" is standard terminology.
> whilst the trains of Cnoid solitons are on an inherently flat
> tesselation of the complex plane.
Yes, except that we are using the real part only. But since cn is
periodic on real arguments, we can regard it as a singly periodic function
on R.
> (OT. The OED definition of tessel@ed is "regularly chequered";
> tessel@ion is "repeated polygons without gaps or overlaps".)
My pseudonym, "T. Essel", thus "tessel@tum.bot", is a juvenile joke (L.
tesselatum, "tiling"), referring in part to "tiling theory" in symbolic
dynamics, which, roughly speaking, treats a huge class of "digitized"
dynamical systems. Examples include "shift spaces", and also "tiling
spaces" such as the space of Penrose tilings. Symbolic dynamical systems
can span the spectrum from "chaotic" (the one dimensional "full shifts"
are classic examples of chaotic dynamical systems) to "highly regular"
("Sturmian shifts" are classic examples of one dimensional dynamical
systems defined on the circle). The latter should be related to
completely integrable Hamiltonian systems. (For example, but Sturmian
shifts and soliton PDEs exhibit one dimensional "quasiperiodicity".) My
interest in solitons was sparked by mysterious remarks by Arnold to the
effect that he knows why "Arnold diffusion" in certain "almost Hamiltonian
systems" gives rise to "spatial patterns" bearing a striking resemblence
to higher dimensional analogues of Penrose rhomb tilings.
> <snip>>The obvious question is probably: have you tried to put your
> >generalized KdV equations into Lax pair form?
> No. In the 90's I did it my way, empirically - ignoring other concepts
> unless I could programme them easily. Nowadays the net drowns me in
> information and I do not know which concepts to test.
I hope to at least explain the Lax formulation and give examples,
including KdV and friends. There are many fascinating connections!
> >Does the Uhlenbeck-Terng loop group formulation help?
> Dunno. Reference please.
The article by Palais is probably the best place to start:
author = {Richard S. Palais},
title = {The Symmetries of Solitons},
journal = {Bull. of the A. M. S.}
volume = {?},
year = {1997}
note = {dg-ga/9708004}}
> P.S. I invite anyone visiting the "Snibston Soliton" to a drink and a
> chat. I live about 8 miles away. E-mail or phone (0509 890511) me in
> advance. R.
I believe the legendary John Baez is currently in Hawkingland. On a
previous visit to Penroseville, he visited Oz, so if you're lucky, maybe
he'll drop in on you! (He knows much more than I do about everything, and
also is much better at explaining what he knows.)
"T. Essel" (hiding somewhere in cyberspace)
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\ntessel@tum.bot writes\n\n>>Oz:\n>> Obviously work on solitons hasn\'t stopped, it just hasn\'t got anywhere.\n>\n>Now I feel like screaming:\n>\n> THE DEVELOPMENT OF SOLITON THEORY IS UNIVERSALLY REGARDED AS ONE OF THE\n> MOST IMPORTANT DEVELOPMENTS IN PHYSICS/APPLIED MATH IN THE PAST 50\n> YEARS! IN SCIENCE, SUCH APPROBATION IS THE EXACT OPPOSITE OF FAILURE!\n>\n>OK, that feels better :-)\n\nIt makes me feel relieved.....\n\n>But let me say it again, more sedately.\n\nA beer helps, I find....\n\n>Soliton theory stands as one of\n>the great achievements (even -the- great achievement) of "nonlinear\n>science" to date. This assessment is in no way diminished by the fact\n>that the theory is perhaps not yet "fully complete" or that deep mysteries\n>like the Painleve conjecture remain unsolved. It is apparently true that\n>the very real and very important successes of soliton theory do not, as\n>yet, include universally recognized deep connections with -particle\n>physics-.\n\nThe fact that schroedinger comes up at all is like someone in gods\nentourage waving a big flag to help us on our way. I\'m particularly\nconcerned that the idea solitons should be of plankian size as an aim. I\ndon\'t see elementary particles as anything but a wave in several\ndimensions. In some dimensions they may be large and low frequency (eg\nelectric field) in others small and fast (eg spatial extent).\n\nI don\'t (when unprobed by energetic probes) see an electron in an s1\norbital as an electron spinning round the proton in a convoluted orbit,\nI see the photon as existing in the whole orbital. That is, the electron\nis really (in this environment) as big as the s1 orbital. This is\nwaaaaay bigger than plankian.\n\n--\nOz\nThis post is worth absolutely nothing and is probably fallacious.\n\nBTOPENWORLD address about to cease. DEMON address no longer in use.\n>>Use oz@farmeroz.port995.com<<\nozacoohdb@despammed.com still functions.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>tessel@tum.bot writes
>>Oz:
>> Obviously work on solitons hasn't stopped, it just hasn't got anywhere.
>
>Now I feel like screaming:
>
> THE DEVELOPMENT OF SOLITON THEORY IS UNIVERSALLY REGARDED AS ONE OF THE
> MOST IMPORTANT DEVELOPMENTS IN PHYSICS/APPLIED MATH IN THE PAST 50
> YEARS! IN SCIENCE, SUCH APPROBATION IS THE EXACT OPPOSITE OF FAILURE!
>
>OK, that feels better :-)
It makes me feel relieved.....
>But let me say it again, more sedately.
A beer helps, I find....
>Soliton theory stands as one of
>the great achievements (even -the- great achievement) of "nonlinear
>science" to date. This assessment is in no way diminished by the fact
>that the theory is perhaps not yet "fully complete" or that deep mysteries
>like the Painleve conjecture remain unsolved. It is apparently true that
>the very real and very important successes of soliton theory do not, as
>yet, include universally recognized deep connections with -particle
>physics-.
The fact that schroedinger comes up at all is like someone in gods
entourage waving a big flag to help us on our way. I'm particularly
concerned that the idea solitons should be of plankian size as an aim. I
don't see elementary particles as anything but a wave in several
dimensions. In some dimensions they may be large and low frequency (eg
electric field) in others small and fast (eg spatial extent).
I don't (when unprobed by energetic probes) see an electron in an s1
orbital as an electron spinning round the proton in a convoluted orbit,
I see the photon as existing in the whole orbital. That is, the electron
is really (in this environment) as big as the s1 orbital. This is
waaaaay bigger than plankian.
--
Oz
This post is worth absolutely nothing and is probably fallacious.
BTOPENWORLD address about to cease. DEMON address no longer in use.
>>Use oz@farmeroz.port995.com<<
ozacoohdb@despammed.com still functions.
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\ntessel@tum.bot writes\n\n>You shouldn\'t be. Recall that solitons are extremely persistent--- they\n>don\'t disperse or diffuse way, unlike single wavecrest solutions of\n>ordinary old wave equations. In particular, we don\'t see anything\n>resembling spontaneous decay of solitons! (Or two solitons somehow\n>combining into one, like two mutually approaching billiard balls in one\n>dimension in the center of mass frame.) Also, when we smash particles in\n>a collider, the collison typically produces a spray of new particles plus\n>massless radiation. But, as we saw, when KdV or SG solitons collide,\n>after collision the solitons emerge unscathed except for a phase change,\n>so nothing like a -spray- of new particles, and there is no "radiation" in\n>sight. This remains true for more than one spatial dimension.\n\nI am not surprised that simple ones behave like this, but I would hope\nthat more complex ones, hopefully with more dimensions, would manage\nthis. At a minimum I would be looking to (3+1)D or 4D plus (at least)\none for the electric field, and I would be surprised if this was enough\nif quarks were to be modelled.\n\nI would be expecting multiple solutions, these would be quantised (else\nwhy bother with solitons) but when properly combined have more than one\nsolution for the combination.\n\nLets just look at how some simple real particles behave.\n\nPhoton: IMHO not solitons, just a maxwell-like wave.\n\nElectron: The simplest soliton-like particle we might look at.\n\nWith these two we only have a limited number of options.\n\n1) Deflection of the electron by an electric field.\nAfter all that\'s all light is, except it oscillates.\nThis is not really a soliton-on-soliton effect, you just have to show\nthe equivalent to a (2D peaked) water soliton on an inclined water\nsurface, will \'fall\' down the slope. Instinctively, I think it should.\n\n2) Deflection. Trickier. None of the web examples show this. They all\njust show them simply \'passing through each other\' (or bouncing off,\nwhich is indistinguishable). Unfortunately they are all (1+1)d solitons\n(ie travelling on a line). To get deflection we would need a (2+1)D\nsoliton (ie moving in a plane). To mimic an electron we also need to\nhave (in one of the dimensions) the core of the soliton (representing\nthe particle) sitting on top of a hump that mimics the electric field of\nthe electron. That way an approaching electro-soliton would have to run\n\'uphill\'.\n\n3) Annihilation. In some ways easier to see following the model in (2).\ne+ will be electric hump down (say) and e- electric hump up. Now I\nforget, but since a gamma cannot produce an e+e- pair unless it can dump\nmomentum into something else, I wouldn\'t expect annihilation to be\nsimply modelled. The best we could so would be to show plausible\ndeflections. An e+e- \'atom\' would be good.....\n\n4) Radiation. Well you ought to be able to get this by taking an e+ and\ngiving it a sinusoidal shaking. Instinctively one can see that this\nshould produce ripples in the (big) electric field \'hump\'.\n\nSo in fact the photon-electron system is limited to deflection as the\nonly new property, maybe some ripples as well.\n\nIf I knew what I was talking about, had the adequate level of\nmathematical understanding and expertise, that\'s where I would be\nlooking.\n\n--\nOz\nThis post is worth absolutely nothing and is probably fallacious.\n\nBTOPENWORLD address about to cease. DEMON address no longer in use.\n>>Use oz@farmeroz.port995.com<<\nozacoohdb@despammed.com still functions.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>tessel@tum.bot writes
>You shouldn't be. Recall that solitons are extremely persistent--- they
>don't disperse or diffuse way, unlike single wavecrest solutions of
>ordinary old wave equations. In particular, we don't see anything
>resembling spontaneous decay of solitons! (Or two solitons somehow
>combining into one, like two mutually approaching billiard balls in one
>dimension in the center of mass frame.) Also, when we smash particles in
>a collider, the collison typically produces a spray of new particles plus
>massless radiation. But, as we saw, when KdV or SG solitons collide,
>after collision the solitons emerge unscathed except for a phase change,
>so nothing like a -spray- of new particles, and there is no "radiation" in
>sight. This remains true for more than one spatial dimension.
I am not surprised that simple ones behave like this, but I would hope
that more complex ones, hopefully with more dimensions, would manage
this. At a minimum I would be looking to (3+1)D or 4D plus (at least)
one for the electric field, and I would be surprised if this was enough
if quarks were to be modelled.
I would be expecting multiple solutions, these would be quantised (else
why bother with solitons) but when properly combined have more than one
solution for the combination.
Lets just look at how some simple real particles behave.
Photon: IMHO not solitons, just a maxwell-like wave.
Electron: The simplest soliton-like particle we might look at.
With these two we only have a limited number of options.
1) Deflection of the electron by an electric field.
After all that's all light is, except it oscillates.
This is not really a soliton-on-soliton effect, you just have to show
the equivalent to a (2D peaked) water soliton on an inclined water
surface, will 'fall' down the slope. Instinctively, I think it should.
2) Deflection. Trickier. None of the web examples show this. They all
just show them simply 'passing through each other' (or bouncing off,
which is indistinguishable). Unfortunately they are all (1+1)d solitons
(ie travelling on a line). To get deflection we would need a (2+1)D
soliton (ie moving in a plane). To mimic an electron we also need to
have (in one of the dimensions) the core of the soliton (representing
the particle) sitting on top of a hump that mimics the electric field of
the electron. That way an approaching electro-soliton would have to run
'uphill'.
3) Annihilation. In some ways easier to see following the model in (2).
e+ will be electric hump down (say) and e- electric hump up. Now I
forget, but since a \gamma cannot produce an e+e- pair unless it can dump
momentum into something else, I wouldn't expect annihilation to be
simply modelled. The best we could so would be to show plausible
deflections. An e+e- 'atom' would be good.....
4) Radiation. Well you ought to be able to get this by taking an e+ and
giving it a sinusoidal shaking. Instinctively one can see that this
should produce ripples in the (big) electric field 'hump'.
So in fact the photon-electron system is limited to deflection as the
only new property, maybe some ripples as well.
If I knew what I was talking about, had the adequate level of
mathematical understanding and expertise, that's where I would be
looking.
--
Oz
This post is worth absolutely nothing and is probably fallacious.
BTOPENWORLD address about to cease. DEMON address no longer in use.
>>Use oz@farmeroz.port995.com<<
ozacoohdb@despammed.com still functions.
Hey, this post is directed to tessel. I couldn't help but notice that you seem to be a
soliton expert, I was hoping you could help me out. I'm looking for an n-soliton
solution to the KdV equation
diff(u (x,t), t) + 6 * u (x,t) * diff(u(x,t), x) + diff( u (x,t), x, x, x ) = 0
We know that solutions of the KdV equation can be systematically obtained from
solutions psi[i] of the free particle Schroedinger equation:
-( diff(psi[i], x, x)) = E[i] * psi[i] via the Wronskian formula
u(x,t) = 2 * diff ( ln (W), x, x) where W = W( psi[1], psi[2], ... , psi[n]) is the
Wronskian determinant composed of psi[ i ]( xi [ i ] )
where xi[ i ] = k[ i ] * (x - 4 * k[ i ]^2 * t) for E[ i ] < 0.
First, we obtain a one-soliton solution.
We consider W = psi( xi ) where psi( xi ) = cosh( xi ) corresponds to a negative energy
E of the Schroedinger equation. xi = k * (x - 4 * k^2 * t)
Then psi = cosh( xi ) = cosh( k * (x - 4 * k^2 * t) ). Computing the determinant of
the 1 x 1 Wronskian matrix we get cosh( k * (x - 4 * k^2 * t) ).
Substituting this into the Wronskian formula given above we obtain:
2 * k^2 * sech^2 ( k * (x - 4 * k^2 * t) ), which is a soluton of the KdV
equation.
Similarly, we can obtain a two-soliton solution if we let
xi[ 1 ] = k[1] * ( x - 4 * k[1]^2 * t) and xi[ 2 ] = k[2] * ( x - 4 * k[2]^2 * t)
then psi[ 1 ] = cosh( xi[ 1 ]) = cosh( k[1] * ( x - 4 * k[1]^2 * t) ) and
psi[ 2 ] = sinh( xi[ 2 ]) = sinh( k[2] * ( x - 4 * k[2]^2 * t) ).
Then the Wronskian matrix is:
[cosh( k[1] * ( x - 4 * k[1]^2 * t) ) sinh( k[2] * ( x - 4 * k[2]^2 * t) )
sinh( k[1] * ( x - 4 * k[1]^2 * t) )*k[1] cosh( k[2] * ( x - 4 * k[2]^2 * t) )*k[2]
Again, by calculating the determinant and substituting this result into the Wronskian
formula we obtain a lengthy solution to the two-soliton equation.
Now my question is: can we repeat the above procedure to obtain an n-soliton
solution? If so, what would the n x n Wronskian matrix look
like, and what would the solution be?
Any help regarding this matter would be greatly appreciated,
thanks
tessel@tum.bot
Aug13-04, 05:41 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Thu, 12 Aug 2004, MICA2 wrote:\n\n<something largely unreadable>\n\nMica, can you please reformat your post with standard length lines?\nWithout the extraneous carraige returns? With the equations formatted by\nhand, if neccessary, to resemble indented equations as in a typeset\ndocument?\n\n(It will greatly improve readability if you can break up lengthy\nexpressions by introducing new quantities, in order to get each equation\nto fit on an indented line! I also like to leave a blank line above and\nbelow any "indented equations". If I plan to cite an equation I may even\nadd a number in parentheses at far right.)\n\nAgain, in happier days the moderators would have had the time to make this\nrequest; -please-, everyone, to reduce the burden on these overworked\nvolunteers--- and to display courtesy for your readers--- take the time to\ncheck for easily corrected problems like this -before- submitting your\npost. TIA!\n\n> I couldn\'t help but notice that you seem to be a soliton expert, I was\n> hoping you could help me out.\n\nNot an expert at all--- I have just been reading some good books/papers\nlately and wanted to share the fun. I might have been able to help you,\nhowever, if I could have read your question! Alas, by the time you could\nreply with your reformatted question (especially given moderation delays),\nI will probably have been taken down for a month or more. Should I\neventually reappear here, and if you are still interested, please ask\nagain.\n\nIf your question was simply: "what is the standard n-soliton solution to\nthe KdV and how can we derive it?", then there are several methods you can\nuse. Hirota\'s "bilinear operator method" is by common agreement the most\nefficient of these. This method (and the application to deriving the\nn-soliton solution) is discussed in the very readable inexpensive\npaperback book\n\nauthor = {P.G. Drazin and R.S. Johnson},\ntitle = {Solitons : an introduction},\npublisher = {Cambridge University Press},\nyear = 1989}\n\nIIRC, this book does not discuss the matrix-expansion way of (in some\nsense) writing down the general n-soliton solution, but IMO this is in any\ncase less useful than an explicit method for constructing them\nrecursively, such as a nonlinear superposition law.\n\nHTH,\n\n"T. Essel" (hiding somewhere in cyberspace)\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Thu, 12 Aug 2004, MICA2 wrote:
<something largely unreadable>
Mica, can you please reformat your post with standard length lines?
Without the extraneous carraige returns? With the equations formatted by
hand, if neccessary, to resemble indented equations as in a typeset
document?
(It will greatly improve readability if you can break up lengthy
expressions by introducing new quantities, in order to get each equation
to fit on an indented line! I also like to leave a blank line above and
below any "indented equations". If I plan to cite an equation I may even
add a number in parentheses at far right.)
Again, in happier days the moderators would have had the time to make this
request; -please-, everyone, to reduce the burden on these overworked
volunteers--- and to display courtesy for your readers--- take the time to
check for easily corrected problems like this -before- submitting your
post. TIA!
> I couldn't help but notice that you seem to be a soliton expert, I was
> hoping you could help me out.
Not an expert at all--- I have just been reading some good books/papers
lately and wanted to share the fun. I might have been able to help you,
however, if I could have read your question! Alas, by the time you could
reply with your reformatted question (especially given moderation delays),
I will probably have been taken down for a month or more. Should I
eventually reappear here, and if you are still interested, please ask
again.
If your question was simply: "what is the standard n-soliton solution to
the KdV and how can we derive it?", then there are several methods you can
use. Hirota's "bilinear operator method" is by common agreement the most
efficient of these. This method (and the application to deriving the
n-soliton solution) is discussed in the very readable inexpensive
paperback book
author = {P.G. Drazin and R.S. Johnson},
title = {Solitons : an introduction},
publisher = {Cambridge University Press},
year = 1989}
IIRC, this book does not discuss the matrix-expansion way of (in some
sense) writing down the general n-soliton solution, but IMO this is in any
case less useful than an explicit method for constructing them
recursively, such as a nonlinear superposition law.
HTH,
"T. Essel" (hiding somewhere in cyberspace)
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