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Local Jet Bundle Formulation of Bäcklund Transformations

With Applications to Non-Linear Evolution Equations

  • F. A. E. Pirani
  • D. C. Robinson
  • W. F. Shadwick

Part of the Mathematical Physics Studies book series (MPST, volume 1)

Table of contents

  1. Front Matter
    Pages I-VIII
  2. F. A. E. Pirani, D. C. Robinson, W. F. Shadwick
    Pages 1-11
  3. F. A. E. Pirani, D. C. Robinson, W. F. Shadwick
    Pages 12-29
  4. F. A. E. Pirani, D. C. Robinson, W. F. Shadwick
    Pages 30-50
  5. F. A. E. Pirani, D. C. Robinson, W. F. Shadwick
    Pages 51-65
  6. F. A. E. Pirani, D. C. Robinson, W. F. Shadwick
    Pages 66-90
  7. F. A. E. Pirani, D. C. Robinson, W. F. Shadwick
    Pages 91-103
  8. F. A. E. Pirani, D. C. Robinson, W. F. Shadwick
    Pages 104-123
  9. Back Matter
    Pages 125-132

About this book

Introduction

The aim of this paper is to show that the theory of jet bundles supplies the appropriate setting for the study of Backlund trans­ formations. These transformations are used to solve certain partial differential equations, particularly non-linear evolution equations. Of course jets have been employed for some time in the theory of partial differential equations, but so far little use has been made of them in applications. In the meanwhile, substantial progress has been made in the study of non-linear evolution equations. This work has been encouraged by the dis­ covery of remarkable properties of some such equations, for example the existence of soliton solutions and of infinite se­ quences of conservation laws. Among the techniques devised to deal with these equations are the inverse scattering method and the Backlund transformation. In our opinion the jet bundle formulation offers a unifying geometrical framework for under­ standing the properties of non-linear evolution equations and the techniques used to deal with them, although we do not consider all of these properties and techniques here. The relevance of the theory of jet bundles lS that it legitimates the practice of regarding the partial derivatives of field variables as independent quantities. Since Backlund trans­ formations require from the outset manipulation of these partial derivatives, and repeated shifts of point of view about which variables are dependent on which, this geometrical setting clari­ fies and simplifies the concepts involved, and offers the prospect of bringing coherence to a variety of disparate results.

Keywords

evolution solution

Authors and affiliations

  • F. A. E. Pirani
    • 1
  • D. C. Robinson
    • 1
  • W. F. Shadwick
    • 1
  1. 1.Department of Mathematics, King’s CollegeUniversity of LondonUK

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-009-9511-6
  • Copyright Information Springer Science+Business Media B.V. 1979
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-90-277-1036-9
  • Online ISBN 978-94-009-9511-6
  • Buy this book on publisher's site