And the point symmetry group is...
Hi, Old Smuggler,
Old Smuggler said:
Yes, except for a couple of typos which I took the liberty to correct.
Oops, I forgot to write the \cot(\theta) term in the Laplacian, didn't I? (That wasn't a trick question, I really did forget, but I am glad to see that you are on the ball!)
Note that I rewrote your B(r,t), V(r,t) as b(r,\theta), v(r,\theta) for convenience in what follows. I took the independent variables to be r, \theta and the dependent variables to be b, v.
The point symmetry group only takes a minute or so to compute via Maple (using the liesymm package to obtain Lie's "determining equations" for the Lie algebra of the point symmetry group). I obtained a three-dimensional Lie algebra whose generators may be given as:
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\vec{X}_1 = b \, \partial_b, \; \vec{X}_2 = r \, \partial_r - v \, \partial_v, \; \vec{X}_3 = \partial_v<br />
In what follows I'll try to address a somewhat broader audience since I expect that symmetry methods should be of wide interest as the most general standard method of attack on systems of nonlinear partial differential equations. These methods are not a magic bullet--- they often don't yield any information you don't already know from more elementary reasoning, and sometimes they don't help at all, e.g. if the symmetry group is trivial. But this is a general attack which is not really very hard to try, so it is usually worth trying as a matter of course.
The general real linear combination of our three generators gives a multiscaling symmetry
r \, \partial_r + k \, b \, \partial_b + (c- v) \, \partial_v
where k, c are arbitrary real constants. From this we can read off a system of ODEs
\dot{r} = r, \; \dot{\theta} = 0, \; \dot{b} = k \, b, \; \dot{v} = c-v
which can be integrated to obtain the transformations in the group
(r, \, \theta, \, b, \, v) \rightarrow (\lambda \, r, \, \theta, \, \lambda^k \, b, \, c+(v-c)/\lambda)
where \lambda > 0 is the parameter of our uniparameter subgroup (for fixed k,c). (I should probably say that the transformations in the point symmetry group are the ones which take the system into a system having the same functional form. Also, the original parameter was the exponent in an exponential factor; for some reason I modified this to give an action by the positive reals under multiplication.) We also obtain the rational invariants of this flow, namely
\theta = c[1], \; b/r^k = c[2], \; (c-v)/r = c[3]
from which we obtain the symmetry Ansatz, which is the point of this exercise:
b(r,\theta) = f(\theta) \, r^k, \; \; v(r,\theta) = c + r \, g(\theta)[/tex]<br />
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Plugging this into the original system of equations, we obtain a system of two ODEs in \theta only. (If this step fails, this indicates a computational error at an earlier stage; elimination is guaranteed if no errors have occured.) One gives f in terms of g; the other is a rather nasty third order nonlinear equation for g with no obvious symmetries. (Since no-one spoke up to tell me a verbatim environment, and since you didn't say whether you are a Maple user, I am not trying to include here the sequence of Maple commands I used, or to give the third order equation I came up with; to avoid transcription errors it is probably best for you to plug in the above Ansatz and find it for yourself.)<br />
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In a special case we obtain the trivial solution<br />
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b(r,\theta) = P_k(\cos(\theta)) \, r^k, \; v(r,\theta) = c<br />
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where P_k is the Legendre polynomial (this solution resembles an "interior mode" in an analogous situation in gtr), but you probably knew that.<br />
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Old Smuggler said:
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I'm glad to hear that an exact solution may perhaps be found without too<br />
much work. Anyway, I appreciate your efforts.
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Well, I'm not very surprised that I got stuck at a hard ODE; this often happens. But fear not, this method is very sensitive to small changes, so transforming to new variables can make all the difference. (In the above, it looks like I used up all the symmetries, so considering other symmetry assumptions probably won't help for the system expressed in the original variables.)<br />
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(By the way, one thing symmetry methods can help you determine is whether or not their exists a transformation to new (independent and dependent) variables in which the new system is linear. I have <i>not</i> tried that criterion in this example.)<br />
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If you often deal with PDEs, I'd strongly encourage you to learn symmetry methods (there are a number of excellent books out now), and to become familiar with Maple or Mathematica packages which automate the most time consuming step, solving the determining equations. These are always linear, so Groebner basis methods can "triangularize" and then solve (by a generalization of Gaussian reduction), and there are Maple and Mathematica routines which carry this out for differential rings. So the bottom line is that the determining equations are always solvable and you can quickly to fairly quickly determine the symmetry group (modulo computational limitations). <br />
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Once you have that, with a bit of experience you can read off a symmetry Ansatz as above and obtain a reduced system. In Lie's approach, each symmetry should enable you to reduce a system of PDEs to a system with one less variable (in a system of ODEs, each symmetry reduces the order by one). These symmetries "combine" in ways determined by the structure of the Lie algebra, but in principle if the symmetry group is sufficiently large you can often reduce to a single ODE and if that is solvable in closed form, you can often obtain complete solutions to the original system in closed form.<br />
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Lie's method is sometimes described as a generalization of "dimensional analysis", and in many cases (as here), the symmetry Ansatz method yields up an Ansatz which represents a multidimensional scaling which could in many cases have been guessed from more elementary reasoning. However, often one does obtain an Ansatz which might not easily have been guessed.<br />
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If there are boundary conditions, you can look at the subgroup of symmetries (acting on the indepedent variables) which preserve the boundary.<br />
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If your system includes some arbitrary functions (e.g. a forcing term) or parameters, you can use "group analysis" to determine special cases for which the symmetry group is enlarged. This is interesting because both because it can lead to solutions and because it can identify useful subcases. In your case, you probably have some "tuneable parameters" in your field equation, so you can apply a symmetry analysis to find out if there are any special values which yield a larger symmetry group.<br />
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In the case of a system which arises from a Lagrangian (as should happen with any "good" classical field theory), Noether's theorem relating variational symmetries (these form a subgroup of the point symmetrics) belongs to the subject of Lie analysis of differential equations, and gives a systematic way to determine conserved quantities (i.e. satisfying divergence identities). Those interested in soliton equations will be intrigued to note that Noether's notions of "generalized symmetries" (the terminology is unfortunately notoriously unstandardized) gives a systematic way to use elaborations of Lie's methods to find an infinite hierarchy of conservation laws.
