Re: Particles from non-linear waves [Was: Solitons...]

tessel_at_tum.bot
Date: 07/25/04 

Date: 25 Jul 2004 09:16:36 -0400

On Fri, 23 Jul 2004, Oz wrote:

Gerard Westendorp <westy31@xs4all.nl>, not wishing to distract me, wrote:

> >As we are studying particle-like wave structures, we can more
> >or less assign a momentum (px, py,..) and an energy (E) to
> >our humps, which should be reasonable localized in space. This
> >makes them kind of solitons, although they may not fit certain
> >definitions of solititons.

Curiously enough, this is what I was just about to try to talk about!
(In the original thread.)

Namely, I want to compute the energy, momentum, and mass of soliton
solutions of the KdV, etc., and to explain how these arise from Noether
analysis of the point symmetry group. But to put this in perspective, I
want to also discuss energy, momentum, etc. for a variety wave equations,
including

1. the one-dimensional wave equation -u_(tt) + u_(xx) = 0

2. the Klein-Gordon equation -u_(tt) + u_(xx) - m^2 u = 0

3. the sine-Gordon equation -u_(tt) + u_(xx) - m^2 sin(u) = 0

4. the linearized KdV -u_t + u_(xxx) = 0

5. the KdV potential -v_t + (v_x)^2/2 + v_(xxx) = 0

6. the MKdV -v_t + (v_x)^3/3 + v_(xxx) = 0

Themes should include symmetries, Noether analysis, finding Lagrangian
formulations of a given PDE by formulating a "potential form" (not always
possible), dispersion relations (especially group versus phase speeds).
One of the points should be that, while solitons have well-defined and
sensible masses and so forth, so does any "isolated traveling wave pulse"
type solutions to other wave equations, so this is -not- a characteristic
property of solitons.

In future I may try to explain properties which -are- characteristic of
solitons, including Baecklund transformations, hierarchies of
Lie-Baecklund symmetries and conservation laws, etc.

Oz commented (fear not, I didn't actually -read- the post, so I'm not much
distracted--- I just skimmed it looking for "momentum"):

> Indeed. One worry I have is that we glibly say 'momentum' without really
> knowing what it is. One day I really *must* do killing vectors

Killing vectors and conformal Killing vectors are a very, very special
case of the point symmetry groups I am talking about. At some point I'd
like to explain that too. Right now it might be confusing vis a vis
possible discussion of how sine-Gordon and Liouville are related to
minimal surfaces in E^3, locally isometric immersions in E^3 of hyperbolic
plane, etc.

> the velocities wrt some 'centre of momentum' (I have doubts whether such
> a thing exists in this non-linear system)

Fear not; a big point will be that this is very general. You might find
this disappointing because this means it is -not- characteristic of
solitons. But I plan to get into things which -are-, eventually.

I should be able to teach you, Oz, to carry out symmetry and Noether
analysis, dispersion analysis, etc., of a wave equation with paper and
pencil, but you mentioned Mathematica. Well, you can easily use existing
very powerful tools in Mathematica and maple for automating these
computations. You don't even need to entirely understand how the
computations are performed in order to get information out of the results,
and in this thread I am actually trying to -avoid- explaining why the
computations work, I am just trying to teach how to perform them. On the
theory that most readers will be much more willing to read (or work out)
the requisite proofs if they already are convinced that they want to
understand this stuff because they've seen its power.

Anyway, I will try to remember to say something about the syntax of the
basic maple commands.

"T. Essel" (hiding somewhere in cyberspace)