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](http://en.wikipedia.org/wiki/Soliton) are very interesting objects for a number of special properties they have, such as being "particle like", being [stable](http://terrytao.wordpress.com/2008/02/19/why-are-solitons-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.



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/>


