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!
 A: 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
$\frac{d}{dt}\frac{\partial L}{\partial\dot{\varphi}_i}-\frac{\partial L}{\partial\varphi_i}$
leading to
$\frac{D}{mgl}\left( \varphi_{i+1} - 2\varphi_{i} + \varphi_{i-1}\right) - \frac{l}{g}\partial_{tt}\varphi_i=\sin\varphi_i$
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
