Solitary waves and their symmetries This is probably a very naive question from a field that I don't have much background from, but a combination of curiosity and the fact that conceptual questions get very good answers here on MO seemed enough motivation to ask it.
Solitons are very interesting objects for a number of special properties they have, such as being "particle like", being stable and conserving their geometric features. Some of these properties are related to symmetries related to the notion of solitons, symmetries whose presence is mysterious to me. 
My question is, therefore, what is a conceptually satisfying explanation of the non-obvious symmetries of solitons? Where do they come from and/or what causes them? I could not even perform a (fruitful) basic search since I don't know even the key-words necessary for it.
 A: 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 :)
A: 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) 
A: 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/
A: One important point to make is that any (multi)soliton solution is invariant under a suitable linear combination of infintesimal higher symmetries of the system under study, cf. e.g. Ch. 5 of the book Applications of Lie groups to differential equations by Peter Olver and Ch. 3 of the book Solitons in Mathematics and Physics by Alan Newell.
