Integrability, Self-Duality, and Twistor Theory [Hardcover]

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.

Part I: Self-Duality And Integrable Equations
1. Mathematical background
2. The self-dual Yang-Mills equations
3. Symmetries and reduction
4. Reductions to three dimensions
5. Reductions to two dimensions
6. Reduction to one dimension
7. Hierarchies
8. Other self-duality equations
Part II: Twistor Theory
9. Mathematical background
10. Twistor space and the ward construction
11. Reductions of the ward construction
12. Generalizations of the twistor construction
13. Boundary conditions
14. Construction of exact solutions
Appendix A. 1 Lifts and invariant connections
Appendix B. 2 Active and passive gauge transformations
Appendix A. 3 The Drinfeld-Sokolov equations

This excellently written book should be of interest to two distinct grounps. Geometrists will be able to learn about integrable systems, and, vice-versa, interable theorists will be introduced to the geometry underlying many of their constructions. --Mathematical Reviews