# Soliton mechanics

How much of classical mechanics can be modelled with solitons?

What I am aware of is that single solitons behave in a way like free particles: they move along as stable entities with constant velocity. But:

• Can solitons interact with an external field?
• Can solitons collide?
• Can solitons interact with each other by general (central) forces?

That means: Are there PDEs with solutions that can be interpreted as

• a particle in an external field
• two colliding particles
• two generically interacting particles

obeying appropriate laws of classical mechanics?

How would the mass/momentum of a soliton come into play?

Pointers to references would be very welcome!

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I think your question might be reformulated in some way. The general approach to solitons is to derive a differential equation with respect to a given potential. Then, the solution to this equation might have soliton solution, which of course interact with the external field, can collide etc.

But I think this may not cover what you are thinking off. Let me give you a quick derivation of Sine-Gordon equation arising from a model of coupled pendulums. This might help to specify what you want to know.

## Sin Gordon Equation: elastic coupled pendulums

The kinetic energy of $N$ pendulums with mass $m$ and length $l$ is given by
$T = \sum_{i=1}^N\frac{m}{2}l^2\dot{\varphi}_i^2$

The potential energy is a little more complicated: $U = \sum_{i=1}^Nmgl\left( 1 - \cos\varphi_i\right) + \sum_{i=1}^{N-1}\frac{D}{2}\left( \varphi_{i+1}-\varphi_{i}\right)^2$
where $D$ accounts for the strength of the band and $g$ for gravitation.

The Lagrange function is then given by $L = T - U$ and we find the equations of motion using Lagrangian mechanics via the famous

$\frac{d}{dt}\frac{\partial L}{\partial\dot{\varphi}_i}-\frac{\partial L}{\partial\varphi_i}$

$\frac{D}{mgl}\left( \varphi_{i+1} - 2\varphi_{i} + \varphi_{i-1}\right) - \frac{l}{g}\partial_{tt}\varphi_i=\sin\varphi_i$

Now, by letting $N\rightarrow\infty$ and rescaling of coordinates we arrive at the famous Sine-Gordon-equation for the continuous system:

$\partial_{\rho\rho}\varphi - \partial_{\tau\tau}\varphi = \sin\varphi$

This equation has mono- or multisoliton solutions which are nicely described in the Wikipedia article mentioned earlier.

You can see that for the derivation of the equation of motion we needed to have the potential already at hand, it does not work the other way round as far as I know. The solutions are very interesting and widely used throughout physics to explain "non-dispersive" water waves, scattering of particles and so on.

## Some further remarks

For a much deeper insights you might have a look at the book "Solitonen - Nichtlineare Strukturen" by R. Meinel written in German (and judging from your name you might understand) or research articles like Non-commutative soliton scattering by Lindström et al.

As far as I know there does not exist a strict kind of approach that will tell you if a system has soliton solutions or not; the search for such solutions is a science on its own. An expert for solitons in optical systems is O. Egorov, some of his papers on the subject can be found on google scholar.

Sincerely

Robert

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Thanks for your elaborate answer! I'll be happy to dive into it. –  Hans Stricker Dec 12 '10 at 15:54
@Robert: what do you need the original particle potential for? Could you elaborate on that? AFAIK non-linear PDE are being studied on their own in mathematics with complete disregard to where they come from; and indeed many equations come from multiple places so it's not quite correct to assume that there is always some sort of "coupled oscillators" working behind the scenes. –  Marek Dec 12 '10 at 16:41
@Marek: You are absolutely right. You may describe several systems with the same PDE. What I wanted to point out was that you have a system, you derive a PDE for it and may find solitons. You cannot take a soliton on its own and ask how it interacts with some potential, you always have to find solutions of the whole system (unless you're working with some perturbation ansatz). I hope I could clearify my point. –  Robert Filter Dec 12 '10 at 16:48
Sine Gordon solitons correspond to elastically coupled pendulums. KdV solitons correspond to free particles. My question was: Can there be in general PDEs with soliton solutions corresponding to arbitrary mechanical systems? Or not in general? Why then? –  Hans Stricker Dec 12 '10 at 22:31
@Hans Stricker: I added some remarks to the answer. Is this along your lines of thought? Sincerely –  Robert Filter Dec 13 '10 at 8:44