Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations

by I.S. Krasil'shchik

Book Overview

To our wives, Masha and Marian Interest in the so-called completely integrable systems with infinite num- ber of degrees of freedom was aroused immediately after publication of the famous series of papers by Gardner, Greene, Kruskal, Miura, and Zabusky [75, 77, 96, 18, 66, 19J (see also [76]) on striking properties of the Korteweg-de Vries (KdV) equation. It soon became clear that systems of such a kind possess a number of characteristic properties, such as infinite series of symmetries and/or conservation laws, inverse scattering problem formulation, L - A pair representation, existence of prolongation structures, etc. And though no satisfactory definition of complete integrability was yet invented, a need of testing a particular system for these properties appeared. Probably one of the most efficient tests of this kind was first proposed by Lenard [19]' who constructed a recursion operator for symmetries of the KdV equation. It was a strange operator, in a sense: being formally integro-differential, its action on the first classical symmetry (x-translation) was well-defined and produced the entire series of higher KdV equations; but applied to the scaling symmetry, it gave expressions containing terms of the type J u dx which had no adequate interpretation in the framework of the existing theories. It is not surprising that P. Olver wrote "The de- duction of the form of the recursion operator (if it exists) requires a certain amount of inspired guesswork. . . " [80, p.

• Format: Paperback
• Language: English
• ISBN: 904815460X
• ISBN13: 9789048154609
• Release Date: December 2010
• Publisher: Springer
• Length: 384 Pages
• Weight: 1.24 lbs.
• Dimensions: 0.8" x 6.1" x 9.2"