Mathematics and Its Applications

Krasil'shchik, I.S.; Kersten, P.H.

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.

Details

Published by: Springer

Publication Date: 2000-05-31

Format: Hardcover

ISBN-13: 9780792363156

DOI: 10.1007/978-94-017-3196-6

Dimensions: 234cm x156cm

Pages: 384