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The Symmetries of Solitons

Author: Richard S. Palais
Journal: Bull. Amer. Math. Soc. 34 (1997), 339-403
MSC (1991): Primary 58F07, 35Q51, 35Q53, and, 35Q55
MathSciNet review: 1462745
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Abstract: In this article we will retrace one of the great mathematical adventures of this century-the discovery of the soliton and the gradual explanation of its remarkable properties in terms of hidden symmetries. We will take an historical approach, starting with a famous numerical experiment carried out by Fermi, Pasta, and Ulam on one of the first electronic computers, and with Zabusky and Kruskal's insightful explanation of the surprising results of that experiment (and of a follow-up experiment of their own) in terms of a new concept they called ``solitons''. Solitons however raised even more questions than they answered. In particular, the evolution equations that govern solitons were found to be Hamiltonian and have infinitely many conserved quantities, pointing to the existence of many non-obvious symmetries. We will cover next the elegant approach to solitons in terms of the Inverse Scattering Transform and Lax Pairs, and finally explain how those ideas led step-by-step to the discovery that Loop Groups, acting by ``Dressing Transformations'', give a conceptually satisfying explanation of the secret soliton symmetries.

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Additional Information

Richard S. Palais
Affiliation: Department of Mathematics, Brandeis University, Waltham, Massachusetts 02254
Address at time of publication: The Institute for Advanced Study, Princeton, New Jersey 08540

Keywords: Solitons, integrable systems, hidden symmetry, Korteweg-de Vries equation, Nonlinear Schr\"{o}dinger equation, Lax pair, Inverse Scattering Transform, loop group
Received by editor(s): May 7, 1997
Received by editor(s) in revised form: August 6, 1997
Additional Notes: During the preparation of this paper, the author was supported in part by the Mathematics Institute and Sonderforschungsbereich 256 of Bonn University.
Article copyright: © Copyright 1997 Richard S. Palais