Solitary waves and their symmetries

I have a rather long expository paper in the Bulletin of the American Math. Society called "The Symmetries of Solitons", in which I try to answer just this question. It is aimed at someone without prior familiarity with the field of soliton mathematics, and as well as developing the mathematical tools needed to understand the subject, it also gives an account of its long and interesting history, starting with the first known observation of a soliton by Lord Russell, and describing the remarkable mathematical experiment of Fermi, Pasta, and Ulam, using one of the first electronic computers, an experiment that gave rise to the remarkable modern development of the subject. The article is freely available here (click on the link "Retrieve article in PDF"):

http://www.ams.org/journals/bull/1997-34-04/S0273-0979-97-00732-5/

Palais' fantastic summary tells most of what is necessary to know if you want to understand solitons (and I re-read that paper with pleasure). I thought I'd add just one more perspective since the more the better.

Solitons are symmetric because they are extremals. They are not minimizers, but you can think of them as one version of minimizers for evolution equations. So now, why extremals are symmetric? This is a very basic phenomenon and not surprising at all. Think of the inequality $a^2+b^2-2ab\ge0$, you get equality when $a=b$ which is the most symmetric configuration. More interestingly, think of Sobolev embeddings $\|u\|_X \le C \|u\|_Y$, (with optimal $C$): in general the inequality is strict and if you get equality the function $u$ must be radial. You can explain this fact in several ways more or less equivalent to each other. One way: if your functional has some symmetry, you have available symmetrization 'moves' which improve the inequality, so if you start with a non symmetric $u$ you can do better by symmetrizing it. For instance, given a nonradial function, if you radially rearrange it you can make the two sides of the Sobolev inequality closer. I hope this is not too vague :)

I think this survey is precisely about your question. (BTW, what I find most astonishing is that p-adic versions of them exist and play a big role)