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 guage transformations


• Appendix A. 3 The Drinfeld-Sokolov equations