Local Jet Bundle Formulation of Bäckland Transformations

1. Front Matter

   Pages I-VIII

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2. Introduction

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

   Pages 1-11

3. Jet Bundles

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

   Pages 12-29

4. Bäcklund Maps : Simplest Case

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

   Pages 30-50

5. Bäcklund Maps : General Case

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

   Pages 51-65

6. Connections

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

   Pages 66-90

7. One Parameter Families of Bäcklund Maps

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

   Pages 91-103

8. Solutions of the Bäcklund Problem

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

   Pages 104-123

9. Back Matter

   Pages 125-132

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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.

• evolution
• solution

• Department of Mathematics, King’s College, University of London, UK

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

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