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Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations

  • I. S. Krasil’ shchik
  • P. H. M. Kersten

Part of the Mathematics and Its Applications book series (MAIA, volume 507)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. I. S. Krasil’ shchik, P. H. M. Kersten
    Pages 1-55
  3. I. S. Krasil’ shchik
    Pages 57-97
  4. I. S. Krasil’ shchik, P. H. M. Kersten
    Pages 99-153
  5. I. S. Krasil’ shchik, P. H. M. Kersten
    Pages 155-185
  6. I. S. Krasil’ shchik, P. H. M. Kersten
    Pages 187-242
  7. I. S. Krasil’ shchik, P. H. M. Kersten
    Pages 243-308
  8. I. S. Krasil’ shchik, P. H. M. Kersten
    Pages 309-348
  9. I. S. Krasil’ shchik, P. H. M. Kersten
    Pages 349-372
  10. Back Matter
    Pages 373-384

About this book

Introduction

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.

Keywords

Mathematica algebra algorithms differential geometry geometry partial differential equation

Authors and affiliations

  • I. S. Krasil’ shchik
    • 1
  • P. H. M. Kersten
    • 2
  1. 1.Independent University of Moscow and Moscow Institute for Municipal EconomyMoscowRussia
  2. 2.Faculty of Mathematical SciencesUniversity of TwenteEnschedeThe Netherlands

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-017-3196-6
  • Copyright Information Springer Science+Business Media B.V. 2000
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-90-481-5460-9
  • Online ISBN 978-94-017-3196-6
  • Buy this book on publisher's site