Integrability, Self-Duality, and Twistor Theory

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Many of the familiar integrable systems of equations are symmetry reductions of self-duality equations on a metric or on a Yang-Mills connection. For example, the Korteweg-de Vries and non-linear Schrodinger equations are reductions of the self-dual Yang-Mills equation. This book explores in detail the connections between self-duality and integrability, and also the application of twistor techniques to integrable systems. It supports two central theories: that the symmetries of self-duality equations provide a natural classification scheme for integrable systems; and that twistor theory provides a uniform geometric framework for the study of Backlund transformations, the inverse scattering method, and other such general constructions of integrability theory. The book will be useful to researchers and graduate students in mathematical physics.

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Product Details

Meet the Author

Mathematical Institute, Oxford

Mathematical Institute, Oxford

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Table of Contents

1 Introduction 1
2 Mathematical background I 13
3 The ASD Yang-Mills equation 32
4 Reduction of the ASDYM equation 43
5 Reduction to three dimensions 59
6 Reduction to two dimensions 67
7 Reductions to one dimension 95
8 Hierarchies 111
9 Mathematical background II 137
10 The twistor correspondence 171
11 Reductions of the Penrose-Ward transform 204
12 Twistor construction of hierarchies 244
13 ASD metrics 284
A Active and passive gauge transformations 317
B The Drinfeld-Sokolov construction 319
C Poisson and symplectic structures 328
D Reductions of the ASDYM equation 339
References 343
A note on notation 356
Index of notation 357
Index 359
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