# 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.

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This is a good question. A quick comment (as I'm off to a meeting) is that you could start by googling Bäcklund transformations. Beware that the wiki page is not particularly interesting, though. –  José Figueroa-O'Farrill Sep 1 '10 at 9:57
If one considers the completely integrable case, then I believe the things called 'symmetry' can be reduced to simple transformations of the spectrum of the underlying Lax operator $L$. For example the existence of a soliton then corresponds to the existence of an eigenvalue of $L$. However, it is not clear to me from your question, if you are interested in this case, or the more 'pde' case discussed in Tao's article. –  Helge Sep 1 '10 at 12:08

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/

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That's a very beautiful and readable article! –  Thomas Riepe Sep 1 '10 at 12:55
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 :)