View Full Version : Re: Solitons in One Post
John Baez
Aug18-04, 04:08 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>\nIn article <40DD7A91.9030406@xs4all.nl>,\nGerard Westendorp <westy31@xs4all.nl> wrote:\n\n>tessel@tum.bot wrote:\n\n>> 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".\n\n>How would this work out for the linear wave equation?\n>\n>My guess:\n>The infinite conserved quantities are just the energies of the\n>linearly independent eigenmodes.\n\nRight - for the linear wave equation you do a Fourier transform\nto break down any solution into plane waves, and the amplitudes\nof these plane waves are a complete set of conserved quantities\nwhose Poisson brackets all vanish. So, this system is "completely\nintegrable" according to my favorite definition of this term.\n\nIn lowbrow terms, the linear wave equation is just a bunch of\nuncoupled harmonic oscillators.\n\n>Actually, I do not quite understand what it means physically to\n>have a non- completely integrable system. Chaos?\n\nThe presence of chaos rules out complete integrability, but I think\nthere are lots of systems that are neither chaotic nor completely\nintegrable. Note however that "chaotic" is a vague term that can\nbe made precise in different ways, so one would have to pick one\nto really dig into this question!\n\n(Complete integrability also has different definitions, but I have\nmy favorite definition - mentioned above.)\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>In article <40DD7A91.9030406@xs4all.nl>,
Gerard Westendorp <westy31@xs4all.nl> wrote:
>tessel@tum.bot wrote:
>> 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".
>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.
Right - for the linear wave equation you do a Fourier transform
to break down any solution into plane waves, and the amplitudes
of these plane waves are a complete set of conserved quantities
whose Poisson brackets all vanish. So, this system is "completely
integrable" according to my favorite definition of this term.
In lowbrow terms, the linear wave equation is just a bunch of
uncoupled harmonic oscillators.
>Actually, I do not quite understand what it means physically to
>have a non- completely integrable system. Chaos?
The presence of chaos rules out complete integrability, but I think
there are lots of systems that are neither chaotic nor completely
integrable. Note however that "chaotic" is a vague term that can
be made precise in different ways, so one would have to pick one
to really dig into this question!
(Complete integrability also has different definitions, but I have
my favorite definition - mentioned above.)
John Baez
Aug18-04, 04:08 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\nIn article <Ibo9u9GmlpzAFwTg@farmeroz.port995.com>,\nOz <oz@farmeroz.port995.com> wrote:\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\nHi!\n\nThat\'s not the *main* reason people are interested in solitons\nthese days - but here are two interesting things:\n\n1) Quantum chromodynamics (QCD) is believed to explain things\nlike how quarks get confined into mesons and baryons. But in\npractice, it\'s such a complicated theory that we need to break\nout our computers and simulate it numerically to see how this\nconfinement works! Luckily, there\'s a very cute simplified\nmodel called the "Skyrme model" in which mesons and baryons\nshow up as solitons! People have fun trying to see how this\nsimplified model shows up as an approximation to QCD.\n\nFor the hardcore physicists out there, the nice thing about\nthe Skyrme model is that it involves a SU(3)-valued field on\nspacetime, where SU(3) is the gauge group of QCD. This model\nadmits topological solitons since pi_3(SU(3)) = Z. Witten\nshowed how to get baryons with fermionic statistics out of\nthe Skyrme model by adding a topological term to the Lagrangian,\nnow called the Wess-Zumino-Witten term.\n\nFor more variations on these ideas, try this:\n\nhttp://www.arXiv.org/hep-ph/0009006\n\n2) In some of the many "dualities" between different string\ntheories that people are studying these days, particles or\nmembranes that are "fundamental" in one description of a theory\nturn out to be solitons in another. Among the top-cited papers\non the physics arXiv is Duff, Khuri and Lu\'s "String Solitons" -\na review article on this idea which helped give birth to dreams\nof "M-theory", a mysterious theory underlying all the different\nsuperstring theories.\n\nHere\'s a nice interview of Duff about this stuff, which you don\'t\nneed a PhD to understand:\n\nhttp://www.esi-topics.com/brane/interviews/MichaelDuff.html\n\nAnd here\'s that paper:\n\nhttp://www.arXiv.org/abs/hep-th/9412184\n\nSo, solitons are alive and well in particle physics!\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>In article <Ibo9u9GmlpzAFwTg@farmeroz.port995.com>,
Oz <oz@farmeroz.port995.com> 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?
Hi!
That's not the *main* reason people are interested in solitons
these days - but here are two interesting things:
1) Quantum chromodynamics (QCD) is believed to explain things
like how quarks get confined into mesons and baryons. But in
practice, it's such a complicated theory that we need to break
out our computers and simulate it numerically to see how this
confinement works! Luckily, there's a very cute simplified
model called the "Skyrme model" in which mesons and baryons
show up as solitons! People have fun trying to see how this
simplified model shows up as an approximation to QCD.
For the hardcore physicists out there, the nice thing about
the Skyrme model is that it involves a SU(3)-valued field on
spacetime, where SU(3) is the gauge group of QCD. This model
admits topological solitons since \pi_3(SU(3)) = Z. Witten
showed how to get baryons with fermionic statistics out of
the Skyrme model by adding a topological term to the Lagrangian,
now called the Wess-Zumino-Witten term.
For more variations on these ideas, try this:
http://www.arXiv.org/http://www.arxiv.org/abs/hep-ph/0009006
2) In some of the many "dualities" between different string
theories that people are studying these days, particles or
membranes that are "fundamental" in one description of a theory
turn out to be solitons in another. Among the top-cited papers
on the physics arXiv is Duff, Khuri and Lu's "String Solitons" -
a review article on this idea which helped give birth to dreams
of "M-theory", a mysterious theory underlying all the different
superstring theories.
Here's a nice interview of Duff about this stuff, which you don't
need a PhD to understand:
http://www.esi-topics.com/brane/interviews/MichaelDuff.html
And here's that paper:
http://www.arXiv.org/abs/http://www.arxiv.org/abs/hep-th/9412184
So, solitons are alive and well in particle physics!
John Baez
Aug18-04, 09:58 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>\nIn article <40ee5c01\\$1@news.sentex.net>, <tessel@tum.bot> wrote:\n\n>On Wed, 7 Jul 2004, Gerard Westendorp wrote:\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\n>Gosh. But these are -deep- water waves, so I don\'t see how they would\n>arise from the KdV right now.\n\nI don\'t know if these huge waves are solitons. It seems unlikely -\nI doubt they go very far. But, I\'ve read papers claiming there are\nsolitons in the Indian ocean.\n\nLet\'s see...\n\nhttp://www.ifm.uni-hamburg.de/ers-sar/Sdata/oceanic/intwaves/andsea/intro/\n\nInternal Solitary Waves in the Andaman Sea\n\nThe Andaman Sea of the Indian Ocean is known to be one of the sites in the\nworld\'s ocean where extraordinarily large internal solitons are encountered.\n\nFor centuries seafarers passing through the Strait of Malacca on their\njourneys between India and the Far East have noticed that in the Andaman\nSea bands of strongly increased surface roughness often occur. These have\nalso been referred to as bands of choppy water or ripplings and have been\nfound mainly between the Nicobar Islands and the north east coast of Sumatra.\nA description of such bands of choppy water observed from ships in the\nwestern approaches of the Malacca Strait can, e.g., be found in the book of\nMauray which was published in 1861 and which is quoted in Osborne and\nBurch (1980): "The ripplings are seen in calm weather approaching from a\ndistance, and in the night their noise is heard a considerable time before\nthey come near. They beat against the sides of a ship with great violence,\nand pass on, the spray sometimes coming on deck; and by carrying out\noceanographic measurements from a ship, a small boat could not always\nresist the turbulence of these remarkable ripplings".\n\nPerry and Schimke (1965) were the first to show by oceanographic\nmeasurements carried out from a ship that these bands of choppy water\nin the Andaman Sea are associated with large-amplitude oceanic internal\nwaves. Later Osborne and Burch (1980) analyzed oceanographic data collected\nby the Exxon Production Research Company in the southern Andaman Sea with\nthe aim to assess the impact of underwater current fluctuations associated\nwith oceanic internal waves on drilling operations carried out from a\ndrill ship. They concluded that the visually observed roughness bands are\ncaused by internal solitons which can be described by the Korteweg -\nde Vries equation (1885).\n\nThe oceanographic measurements of Osborne and Burch (1980) showed that the\ntime interval between the first solitons in the packets was typically\n40 minutes and then decreased towards the end. In one event, the amplitude\n(peak-to-trough distance) of the foremost soliton was estimated to be 60 m,\ni.e., warm water from above was pushed down by the internal soliton by 60 m.\nThe roughness bands associated with one of the internal soliton packets\nextended from horizon to horizon and were 600 to 1200 m wide. The first\nband of choppy water consisted of breaking waves about 1.8 m high. The\nbackground wave field preceding this band had only a waveheight of 0.6m.\nBehind this band of strongly increased surface roughness, the waveheight\ngradually decreased and a band of reduced surface roughness followed,\nwhich had a waveheight of less than 0.1 m and looked "as smooth as a\nmillpond".\n\nThe key thing here is that the surface waves are just a sideffect of\n"internal waves" - waves occuring between ocean layers of different\ntemperature/salinity and thus different density. Apparently it\'s these\ninternal waves that are described by the KdV equation and can form\nsolitons.\n\nA bit more:\n\nhttp://www.ifm.uni-hamburg.de/ers-sar/Sdata/oceanic/intwaves/intro/index.html\n\nOceanic internal waves\n\nInternal waves are waves in the interior of the ocean. They exist when the\nwater body consists of layers of different density. This difference in\nwater density is mostly due to a difference in water temperature, but\ncan also be due to a difference in salinity. Often the density structure\nof the ocean can be approximated by two layers. The interface between\nlayers of different densities is called pycnoline. When the density\ndifference is due to temperature it is called thermocline, and when\nit is due to salinity it is called halocline.\n\n[...]\n\nInternal waves in the ocean typically have wavelengths from hundreds of\nmeters to tens of kilometers and periods from tens of minutes to several\nhours. Their amplitude (peak-to trough distance) often exceeds 50 m.\n\n[...]\n\nThe tidally generated internal waves are usually highly nonlinear and\noccur often in wave packets. The distance between the waves in a wave\npacket and also the amplitude decrease from the front to the back.\nThe amplitude of large internal waves can exceed 50 m in some cases.\nTheoretically, these highly nonlinear waves are often described in\nterms of internal solitons. Thus a wave packet consists of several\nsolitons. Since soliton theory was developed by Korteweg and De Vries\n(1895), hundreds of papers have been published dealing with this subject.\nSoliton theories applicable to the description of the generation and\npropagation of internal solitary waves predict that, if the depth of\nthe upper water layer is much smaller than the depth of the lower layer,\nthen the internal solitary wave must be a "wave of depression". This\nmeans that this soliton is associated with a depression of the pycnocline\nas depicted schematically in Fig. 2 and as measured in the ocean (see\nFig. 3).\n\nCool stuff!\n\n(I\'ll cc this to you in case you didn\'t notice that I\'ve been posting\nup a storm - pardon the pun - both on this thread and on a thread or\ntwo entitled "What is Complete Integrability?".)\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>In article <40ee5c01$1@news.sentex.net>, <tessel@tum.bot> wrote:
>On Wed, 7 Jul 2004, Gerard Westendorp wrote:
>> 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.
I don't know if these huge waves are solitons. It seems unlikely -
I doubt they go very far. But, I've read papers claiming there are
solitons in the Indian ocean.
Let's see...
http://www.ifm.uni-hamburg.de/ers-sar/Sdata/oceanic/intwaves/andsea/intro/
Internal Solitary Waves in the Andaman Sea
The Andaman Sea of the Indian Ocean is known to be one of the sites in the
world's ocean where extraordinarily large internal solitons are encountered.
For centuries seafarers passing through the Strait of Malacca on their
journeys between India and the Far East have noticed that in the Andaman
Sea bands of strongly increased surface roughness often occur. These have
also been referred to as bands of choppy water or ripplings and have been
found mainly between the Nicobar Islands and the north east coast of Sumatra.
A description of such bands of choppy water observed from ships in the
western approaches of the Malacca Strait can, e.g., be found in the book of
Mauray which was published in 1861 and which is quoted in Osborne and
Burch (1980): "The ripplings are seen in calm weather approaching from a
distance, and in the night their noise is heard a considerable time before
they come near. They beat against the sides of a ship with great violence,
and pass on, the spray sometimes coming on deck; and by carrying out
oceanographic measurements from a ship, a small boat could not always
resist the turbulence of these remarkable ripplings".
Perry and Schimke (1965) were the first to show by oceanographic
measurements carried out from a ship that these bands of choppy water
in the Andaman Sea are associated with large-amplitude oceanic internal
waves. Later Osborne and Burch (1980) analyzed oceanographic data collected
by the Exxon Production Research Company in the southern Andaman Sea with
the aim to assess the impact of underwater current fluctuations associated
with oceanic internal waves on drilling operations carried out from a
drill ship. They concluded that the visually observed roughness bands are
caused by internal solitons which can be described by the Korteweg -
de Vries equation (1885).
The oceanographic measurements of Osborne and Burch (1980) showed that the
time interval between the first solitons in the packets was typically
40 minutes and then decreased towards the end. In one event, the amplitude
(peak-to-trough distance) of the foremost soliton was estimated to be 60 m,
i.e., warm water from above was pushed down by the internal soliton by 60 m.
The roughness bands associated with one of the internal soliton packets
extended from horizon to horizon and were 600 to 1200 m wide. The first
band of choppy water consisted of breaking waves about 1.8 m high. The
background wave field preceding this band had only a waveheight of .6m.
Behind this band of strongly increased surface roughness, the waveheight
gradually decreased and a band of reduced surface roughness followed,
which had a waveheight of less than .1 m and looked "as smooth as a
millpond".
The key thing here is that the surface waves are just a sideffect of
"internal waves" - waves occuring between ocean layers of different
temperature/salinity and thus different density. Apparently it's these
internal waves that are described by the KdV equation and can form
solitons.
A bit more:
http://www.ifm.uni-hamburg.de/ers-sar/Sdata/oceanic/intwaves/intro/index.html
Oceanic internal waves
Internal waves are waves in the interior of the ocean. They exist when the
water body consists of layers of different density. This difference in
water density is mostly due to a difference in water temperature, but
can also be due to a difference in salinity. Often the density structure
of the ocean can be approximated by two layers. The interface between
layers of different densities is called pycnoline. When the density
difference is due to temperature it is called thermocline, and when
it is due to salinity it is called halocline.
[...]
Internal waves in the ocean typically have wavelengths from hundreds of
meters to tens of kilometers and periods from tens of minutes to several
hours. Their amplitude (peak-to trough distance) often exceeds 50 m.
[...]
The tidally generated internal waves are usually highly nonlinear and
occur often in wave packets. The distance between the waves in a wave
packet and also the amplitude decrease from the front to the back.
The amplitude of large internal waves can exceed 50 m in some cases.
Theoretically, these highly nonlinear waves are often described in
terms of internal solitons. Thus a wave packet consists of several
solitons. Since soliton theory was developed by Korteweg and De Vries
(1895), hundreds of papers have been published dealing with this subject.
Soliton theories applicable to the description of the generation and
propagation of internal solitary waves predict that, if the depth of
the upper water layer is much smaller than the depth of the lower layer,
then the internal solitary wave must be a "wave of depression". This
means that this soliton is associated with a depression of the pycnocline
as depicted schematically in Fig. 2 and as measured in the ocean (see
Fig. 3).
Cool stuff!
(I'll cc this to you in case you didn't notice that I've been posting
up a storm - pardon the pun - both on this thread and on a thread or
two entitled "What is Complete Integrability?".)
John Baez
Aug18-04, 10:34 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>In article <CfP\\$fQBGUs8AFwbq@farmeroz.port995.com>,\nOz <oz@farmeroz.port995.com> wrote:\n\n>Has anyone produced solitons from (1+3)D equations?\n\nYes - for example the Skyrme model of mesons and baryons which\nI mentioned in a post that I sent off last night. However,\nthese are topological solitons, which are a bit less exciting\nthan examples like Chris was mentioning, where there are infinitely\nmany conserved quantities.\n\n(I\'m catching up on reading s.p.r. as the summer winds to a lazy close.)\n\n>In passing I would like to say that your post, in its enthusiasm and\n>range, are verging on the baezesque.\n\nI\'ll take that as a high compliment.\n\n>What puzzles me is why research into solitons and their possible use for\n>explaining particles seemed to have lapsed.\n\nIt hasn\'t lapsed at all! In my post last night I mentioned not\nonly the Skyrme model but also a vast amount of work on "string\ndualities" based on solitons. There\'s a lot of other stuff too.\n\nIf you\'re fond of solitons you should also learn a little about\ninstantons. It only takes a short time. :-)\n\n>I seem to remember reading (in the distant past) that some solitons at\n>least are constrained to a specific amplitude(s), and that an arbitrary\n>pulse would \'decompose\' into a selection of stable \'soliton modes\'. Have\n>you come across this in your wideranging travels of the literature?\n\nI think something like this happens with the KdV equation...\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>In article <CfP$fQBGUs8AFwbq@farmeroz.port995.com>,
Oz <oz@farmeroz.port995.com> wrote:
>Has anyone produced solitons from (1+3)D equations?
Yes - for example the Skyrme model of mesons and baryons which
I mentioned in a post that I sent off last night. However,
these are topological solitons, which are a bit less exciting
than examples like Chris was mentioning, where there are infinitely
many conserved quantities.
(I'm catching up on reading s.p.r. as the summer winds to a lazy close.)
>In passing I would like to say that your post, in its enthusiasm and
>range, are verging on the baezesque.
I'll take that as a high compliment.
>What puzzles me is why research into solitons and their possible use for
>explaining particles seemed to have lapsed.
It hasn't lapsed at all! In my post last night I mentioned not
only the Skyrme model but also a vast amount of work on "string
dualities" based on solitons. There's a lot of other stuff too.
If you're fond of solitons you should also learn a little about
instantons. It only takes a short time. :-)
>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?
I think something like this happens with the KdV equation...
John Baez
Aug18-04, 10:50 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>\nIn article <40f3913d\\$1@news.sentex.net>, <tessel@tum.bot> wrote:\n\n>Interestingly enough, I have seen some comments to the effect that soliton\n>equations seem to like two variables x,t.\n\nThis is because the magical math of loop groups and conformal\nfield theory is special to 2 dimensions - ultimately because\nthe COMPLEX NUMBERS are 2-dimensional. It\'s this magical math\nthat underlies and unites string theory and most exactly solvable\nclassical field theories, quantum field theories, and statistical\nmechanics models in 2 dimensions.\n\nThere\'s other magical math that works in other dimensions, but it\'s\nof a different flavor. For example, only in 4 dimensions does the\nconcept of self-dual curvature 2-form make sense, for reasons closely\nlinked to the 4-dimensionality of the QUATERNIONS. So, this dimension\nis dominated by self-dual Yang-Mills equations, which have "instanton\nsolutions" that can be obtained by Atiyah-Drinfeld-Hitchin-Manin\nmethod - an offshoot of work started by Penrose. These equations\nhave spinoffs in all directions. For example, in topology they lead\nultimately to the fact that R^4 has uncountably many exotic smooth\nstructures, unlike any other R^n!\n\nIn fact I\'ve heard that the self-dual Yang-Mills equations stand\nbehind and unify pretty much all the different 2d completely\nintegrable wave equations you\'ve been explaining, like the\nKorteweg-deVries equations, the nonlinear Schroedinger equation,\nthe sine-Gordon equation, etc. etc. - as well as the smaller list\nof 3d completely integrable wave equations, like the Kadmotsev-Peviashvili\nequations.\n\nAlas, I don\'t understand this AT ALL! It sounds cool, but whenever\nI try to learn about it I sink under the weight of what seem like\nhundreds of cunningly related equations without an overarching principle\nto help me tell the signal from the noise. As someone said in\na seminar about this stuff which I went to as a postdoc:\n\nIt seems like a wonderful merry-go-round, but how do you get on?\n\nSo I prefer to study subjects where I feel I already hold some of\nthe secret keys. Life is too short to explore all the tantalizing\nmysteries out there... sigh.\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>In article <40f3913d$1@news.sentex.net>, <tessel@tum.bot> wrote:
>Interestingly enough, I have seen some comments to the effect that soliton
>equations seem to like two variables x,t.
This is because the magical math of loop groups and conformal
field theory is special to 2 dimensions - ultimately because
the COMPLEX NUMBERS are 2-dimensional. It's this magical math
that underlies and unites string theory and most exactly solvable
classical field theories, quantum field theories, and statistical
mechanics models in 2 dimensions.
There's other magical math that works in other dimensions, but it's
of a different flavor. For example, only in 4 dimensions does the
concept of self-dual curvature 2-form make sense, for reasons closely
linked to the 4-dimensionality of the QUATERNIONS. So, this dimension
is dominated by self-dual Yang-Mills equations, which have "instanton
solutions" that can be obtained by Atiyah-Drinfeld-Hitchin-Manin
method - an offshoot of work started by Penrose. These equations
have spinoffs in all directions. For example, in topology they lead
ultimately to the fact that R^4 has uncountably many exotic smooth
structures, unlike any other R^n!
In fact I've heard that the self-dual Yang-Mills equations stand
behind and unify pretty much all the different 2d completely
integrable wave equations you've been explaining, like the
Korteweg-deVries equations, the nonlinear Schroedinger equation,
the sine-Gordon equation, etc. etc. - as well as the smaller list
of 3d completely integrable wave equations, like the Kadmotsev-Peviashvili
equations.
Alas, I don't understand this AT ALL! It sounds cool, but whenever
I try to learn about it I sink under the weight of what seem like
hundreds of cunningly related equations without an overarching principle
to help me tell the signal from the noise. As someone said in
a seminar about this stuff which I went to as a postdoc:
It seems like a wonderful merry-go-round, but how do you get on?
So I prefer to study subjects where I feel I already hold some of
the secret keys. Life is too short to explore all the tantalizing
mysteries out there... sigh.
Urs Schreiber
Aug18-04, 11:15 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"John Baez" <baez@galaxy.ucr.edu> schrieb im Newsbeitrag\nnews:cfvtkf\\$5m9\\$1@glue.ucr.edu... \n>\n> In article <40f3913d\\$1@news.sentex.net>, <tessel@tum.bot> wrote:\n>\n> >Interestingly enough, I have seen some comments to the effect that\nsoliton\n> >equations seem to like two variables x,t.\n>\n> This is because the magical math of loop groups and conformal\n> field theory is special to 2 dimensions - ultimately because\n> the COMPLEX NUMBERS are 2-dimensional. It\'s this magical math\n> that underlies and unites string theory and most exactly solvable\n> classical field theories, quantum field theories, and statistical\n> mechanics models in 2 dimensions.\n\nThe fact that integrable systems in 2d are prominent can also be seen to be\nrelated to the fact that the conserved quantities of these systems are\nactually, as is elucidated by the method of Lax pairs, holonomies of a\ncertain connection around \'spatial\' curves.\n\nThis means that generalizing this Lax-like construction of conserved\nquantities to higher dimensions should involve higher-order holonomies like\nsurface holonomies, volume holonomies, etc. in the construction of conserved\ncharges.\n\nThere is an interesting paper which discusses the construction of such\nhigher-order holonomies as conserved charges in integrable systems, that\'s\n\nOrlando Alvarez, Luiz A. Ferreira, J. Sanchez Guillen:\nA New Approach to Integrable Theories in any Dimension,\nhep-th/9710147 .\n\nIn particular, they consider the next-best case of 2+1d integrable field\ntheories where the conserved charges are surface holonomies, i.e. holonomies\nin loop space.\n\nIt turns out that for these surface holonomies in loop space to be well\ndefined the connection on loop space has to be flat. Of course this\nharmonizes well with the Lax method, but it is true more generally.\n\nIn fact, the condition in 2-group theory which you founded and which says\nthat the surface group label must equal the holonomy around the surface\n(emphasized by Girelli&Pfeiffer in hep-th/0309173) translates in the\ncontinuum to the statement that\n\nB = -F\n\n(where B is the 2-form and F the field strength of the 1-form on target\nspace)\n\nand then in loop space to the statement that the connection on loop space is\n_flat_, and furthermore of such a form that its form is preserved under\ngauge transformations on loop space.\n\nBut there is a further condition for holonomies in loop space to give the\nsame surface holonomy in target space as obtained by 2-group computations.\nNamely all loops in the foliation of the surface have to share a common\npoint.\n\nAt least this is what I seem to have found today, as I try to outline here:\n\nhttp://golem.ph.utexas.edu/string/archives/000416.html .\n\nI had some correspondence with the authors of the above paper. In fact they\nreally considered loop space connections with B _not_ equal to -F that are\nhowever still flat. This works and does not violate the 2-group consistency\ncondition B = -F because the B they use is really "quasi-abelian" namely\neither an element of an abelian ideal or covariantly constant (in which case\nthe non-abelian factor can be divided out in a sense). See\n\nhttp://golem.ph.utexas.edu/string/archives/000405.html\n\nfor more details.\n\nSo now we are wondering if the flat loop space connection that I considered\nand which do have F = -B for honest non-abelian B show up in the\nconstruction of any conserved charges in 1+2d integrable field theories.\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>"John Baez" <baez@galaxy.ucr.edu> schrieb im Newsbeitrag
news:cfvtkf$5m9$1@glue.ucr.edu...
>
> In article <40f3913d$1@news.sentex.net>, <tessel@tum.bot> wrote:
>
> >Interestingly enough, I have seen some comments to the effect that
soliton
> >equations seem to like two variables x,t.
>
> This is because the magical math of loop groups and conformal
> field theory is special to 2 dimensions - ultimately because
> the COMPLEX NUMBERS are 2-dimensional. It's this magical math
> that underlies and unites string theory and most exactly solvable
> classical field theories, quantum field theories, and statistical
> mechanics models in 2 dimensions.
The fact that integrable systems in 2d are prominent can also be seen to be
related to the fact that the conserved quantities of these systems are
actually, as is elucidated by the method of Lax pairs, holonomies of a
certain connection around 'spatial' curves.
This means that generalizing this Lax-like construction of conserved
quantities to higher dimensions should involve higher-order holonomies like
surface holonomies, volume holonomies, etc. in the construction of conserved
charges.
There is an interesting paper which discusses the construction of such
higher-order holonomies as conserved charges in integrable systems, that's
Orlando Alvarez, Luiz A. Ferreira, J. Sanchez Guillen:
A New Approach to Integrable Theories in any Dimension,
http://www.arxiv.org/abs/hep-th/9710147 .
In particular, they consider the next-best case of 2+1d integrable field
theories where the conserved charges are surface holonomies, i.e. holonomies
in loop space.
It turns out that for these surface holonomies in loop space to be well
defined the connection on loop space has to be flat. Of course this
harmonizes well with the Lax method, but it is true more generally.
In fact, the condition in 2-group theory which you founded and which says
that the surface group label must equal the holonomy around the surface
(emphasized by Girelli&Pfeiffer in http://www.arxiv.org/abs/hep-th/0309173) translates in the
continuum to the statement that
B = -F
(where B is the 2-form and F the field strength of the 1-form on target
space)
and then in loop space to the statement that the connection on loop space is
_flat_, and furthermore of such a form that its form is preserved under
gauge transformations on loop space.
But there is a further condition for holonomies in loop space to give the
same surface holonomy in target space as obtained by 2-group computations.
Namely all loops in the foliation of the surface have to share a common
point.
At least this is what I seem to have found today, as I try to outline here:
http://golem.ph.utexas.edu/string/archives/000416.html .
I had some correspondence with the authors of the above paper. In fact they
really considered loop space connections with B _not_ equal to -F that are
however still flat. This works and does not violate the 2-group consistency
condition B = -F because the B they use is really "quasi-abelian" namely
either an element of an abelian ideal or covariantly constant (in which case
the non-abelian factor can be divided out in a sense). See
http://golem.ph.utexas.edu/string/archives/000405.html
for more details.
So now we are wondering if the flat loop space connection that I considered
and which do have F = -B for honest non-abelian B show up in the
construction of any conserved charges in 1+2d integrable field theories.
John Baez
Aug18-04, 11:15 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>\nIn article <40f646e5\\$1@news.sentex.net>, <tessel@tum.bot> wrote:\n\n>On Tue, 13 Jul 2004, Robert Shaw wrote:\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\n>A very good question. Your expectation that understanding the symmetry\n>group should hold the key to ultimate understanding of any mystery is\n>fully in keeping with the point of view advocated in the review paper by\n>Palais, so you might want to email him to see he suggests.\n\nI\'ve heard that for all these 1+1-dimensional completely integrable\nwave equations the symmetry group is a "loop group" - that is, a\ngroup whose elements are maps\n\nf: S^1 -> G\n\nwhere G is some Lie group, where we multiply these maps pointwise.\n\nYou\'re right - symmetry *is* the key to many mysteries. So,\nthis is where I\'d start if I wanted to understand this stuff better...\nwhich you are making me want to do. Unfortunately I have many\nthings to do.\n\nHmm, this might be good:\n\nSegal, G., Wilson, G., Loop groups and equations of KdV type, Inst.\nHautes Etudes Sci. Publ. Math., 61 (1985), 5-65.\n\nI wish someone would read this and summarize it... in one post!\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>In article <40f646e5$1@news.sentex.net>, <tessel@tum.bot> wrote:
>On Tue, 13 Jul 2004, Robert Shaw wrote:
>> 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've heard that for all these 1+1-dimensional completely integrable
wave equations the symmetry group is a "loop group" - that is, a
group whose elements are maps
f: S^1 -> G
where G is some Lie group, where we multiply these maps pointwise.
You're right - symmetry *is* the key to many mysteries. So,
this is where I'd start if I wanted to understand this stuff better...
which you are making me want to do. Unfortunately I have many
things to do.
Hmm, this might be good:
Segal, G., Wilson, G., Loop groups and equations of KdV type, Inst.
Hautes Etudes Sci. Publ. Math., 61 (1985), 5-65.
I wish someone would read this and summarize it... in one post!
<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>John Baez <baez@galaxy.ucr.edu> writes\n>Right - for the linear wave equation you do a Fourier transform\n>to break down any solution into plane waves, and the amplitudes\n>of these plane waves are a complete set of conserved quantities\n>whose Poisson brackets all vanish. So, this system is "completely\n>integrable" according to my favorite definition of this term.\n>\n>In lowbrow terms, the linear wave equation is just a bunch of\n>uncoupled harmonic oscillators.\n\nYou will be well aware that I am hanging on by my fingernails barely\nfollowing anything at all, but by golly its fascinating. Even better the\nexperts humour me from time to time with a downmarket explanation.\n\nI seem to be rather out on a limb (novel, I know) in that I would like\nan electron (say) to be describable as a soliton. That is the whole\nthing, not some elemental object that we sum to total an electron. The\nproblem is that I am not quite sure what an electron looks like in\n(3+1)D. Well, actually we need another dimension for the electric field,\ntoo.\n\nNaively (which I do ever so well) I would start with a \'stationary\'\nelectron. Now of course here I fall straight over because we don\'t know\nwhere it is in this case, so its quite hard to give it a spatial extent.\nIn a way its dispersing as described by (some of) tessel\'s soliton\nsolutions. OTOH we could localise it by measuring the electric field,\nwhich would disturb it....\n\nTo get round this conundrum I tend to see it as a static electric field\nin the electric dimension and a decaying pancake in the spatial ones\n(IYSWIM).\n\nTo get a peek at its behaviour in the time direction we need to look\nround the side and expose the time extent. This is most easily done by\ngiving oneself a bit of a boost. When we do this we see a periodic\nbehaviour. The electron diffracts and you once said, many moons ago,\nthat this bit looks like waves travelling a lightspeed and attenuating\naway from the particle. Now the bigger the boost the more we look at the\ntime element and the faster go the oscillations but the smaller\n(spatially) the damn thing looks. At infinite boost it would be\noscillating infinitely fast, which is a nonsense. There is however a\nhint at something else going on. Rather than dispersing (as the static\nsituation suggests) we are getting a smaller and more locatable\nparticle. In a sense the particle is dispersing spatially but collapsing\ntemporally. I rather like that concept because it suggests a balance,\nand if we want a particle where all are the same and we wish to model\nthem as solitons then some sort of balancing act will be required.\n\nNB. The \'infinite frequency\' I mentioned above is a problem. However I\nthink it becomes tractable if we move to another description of\ndistance. Its an odd thing that for any given photon all observers will\nagree that the number of wavelengths it takes for it to move between two\nevents (considering it as a particulate photon). Without actually\nworking it out I would expect that to be true of a particle like an\nelectron. This measure is boost-independent. As we (the observer) boost\nto lightspeed this measure stays constant, I just wish I could put my\nfinger on what its trying to say.\n\nEr, I hope there is enough physics here to get round the \'overly\nspeculative\' moderators directive....\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>John Baez <baez@galaxy.ucr.edu> writes
>Right - for the linear wave equation you do a Fourier transform
>to break down any solution into plane waves, and the amplitudes
>of these plane waves are a complete set of conserved quantities
>whose Poisson brackets all vanish. So, this system is "completely
>integrable" according to my favorite definition of this term.
>
>In lowbrow terms, the linear wave equation is just a bunch of
>uncoupled harmonic oscillators.
You will be well aware that I am hanging on by my fingernails barely
following anything at all, but by golly its fascinating. Even better the
experts humour me from time to time with a downmarket explanation.
I seem to be rather out on a limb (novel, I know) in that I would like
an electron (say) to be describable as a soliton. That is the whole
thing, not some elemental object that we sum to total an electron. The
problem is that I am not quite sure what an electron looks like in
(3+1)D. Well, actually we need another dimension for the electric field,
too.
Naively (which I do ever so well) I would start with a 'stationary'
electron. Now of course here I fall straight over because we don't know
where it is in this case, so its quite hard to give it a spatial extent.
In a way its dispersing as described by (some of) tessel's soliton
solutions. OTOH we could localise it by measuring the electric field,
which would disturb it....
To get round this conundrum I tend to see it as a static electric field
in the electric dimension and a decaying pancake in the spatial ones
(IYSWIM).
To get a peek at its behaviour in the time direction we need to look
round the side and expose the time extent. This is most easily done by
giving oneself a bit of a boost. When we do this we see a periodic
behaviour. The electron diffracts and you once said, many moons ago,
that this bit looks like waves travelling a lightspeed and attenuating
away from the particle. Now the bigger the boost the more we look at the
time element and the faster go the oscillations but the smaller
(spatially) the damn thing looks. At infinite boost it would be
oscillating infinitely fast, which is a nonsense. There is however a
hint at something else going on. Rather than dispersing (as the static
situation suggests) we are getting a smaller and more locatable
particle. In a sense the particle is dispersing spatially but collapsing
temporally. I rather like that concept because it suggests a balance,
and if we want a particle where all are the same and we wish to model
them as solitons then some sort of balancing act will be required.
NB. The 'infinite frequency' I mentioned above is a problem. However I
think it becomes tractable if we move to another description of
distance. Its an odd thing that for any given photon all observers will
agree that the number of wavelengths it takes for it to move between two
events (considering it as a particulate photon). Without actually
working it out I would expect that to be true of a particle like an
electron. This measure is boost-independent. As we (the observer) boost
to lightspeed this measure stays constant, I just wish I could put my
finger on what its trying to say.
Er, I hope there is enough physics here to get round the 'overly
speculative' moderators directive....
--
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.
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