SOLITONS, INSTANTONS, AND TWISTORS.

Most nonlinear differential equations arising in natural sciences admit chaotic behaviour and cannot be solved analytically. Integrable systems lie on the other extreme. They possess regular, stable, and well-behaved solutions known as solitons and instantons. These solutions play important roles in pure and applied mathematics as well as in theoretical physics where they describe configurations topologically different from vacuum. While integrable equations in lower space-time dimensions can be solved using the inverse scattering transform, the higher-dimensional examples of anti-self-dual Yang-Mills and Einstein equations require twistor theory. Both techniques rely on an ability to represent nonlinear equations as compatibility conditions for overdetermined systems of linear differential equations. The book provides a self-contained and accessible introduction to the subject. It starts with an introduction to integrability of ordinary and partial differential equations. Subsequent chapters explore symmetry analysis, gauge theory, vortices, gravitational instantons, twistor transforms, and anti-self-duality equations. The three appendices cover basic differential geometry, complex manifold theory, and the exterior differential system. New to this Edition: - contains a new chapter on Abelian vortices, with the emphasis on the integrable cases - includes a derivation of the master dispersionless integrable system underling the general anti-self-dual conformal structures in four dimensions, as well as a discussion of the non-relativistic limit of twistor theory - new examples have been added and errors have been corrected 1:Integrability in classical mechanics 2:Soliton equations and the inverse scattering transform 3:Hamiltonian formalism and zero-curvature representation 4:Lie symmetries and reductions 5:Lagrangian formalism and field theory 6:Gauge field theory 7:Integrability of ASDYM and twistor theory 8:Symmetry reductions and the integrable chiral model 9:Vortices 10:Gravitational instantons 11:Anti-self-dual conformal structures Appendix A: Manifolds and Topology Appendix B: Complex Analysis Appendix C: Overdetermined PDEs