In the last ten to fifteen years there have been many important developments in the theory of integrable equations. This period is marked in particular by the strong impact of soliton theory in many diverse areas of mathematics and physics; for example, algebraic geometry (the solution of the Schottky problem), group theory (the discovery of quantum groups), topology (the connection of Jones polynomials with integrable models), and quantum gravity (the connection of the KdV with matrix models). This is the first book to present a comprehensive overview of these developments. Numbered among the authors are many of the most prominent researchers in the field.

Inhalt

The Inverse Scattering Transform on the Line.- C-Integrable Nonlinear Partial Differential Equations.- Integrable Lattice Equations.- The Inverse Spectral Method on the Plane.- Dispersion Relations for Nonlinear Waves and the Schottky Problem.- The Isomonodromy Method and the Painlevé Equations.- The Cauchy Problem for Doubly Periodic Solutions of KP-H Equation.- Integrable Singular Integral Evolution Equations.- Long-Time Asymptotics for Integrable Nonlinear Wave Equations.- The Generation and Propagation of Oscillations in Dispersive Initial Value Problems and Their Limiting Behavior.- Differential Geometry Hydrodynamics of Soliton Lattices.- Bi-Hamiltonian Structures and Integrability.- On the Symmetries of Integrable Systems.- The n-Component KP Hierarchy and Representation Theory.- Compatible Brackets in Hamiltonian Mechanics.- Symmetries - Test of Integrability.- Conservation and Scattering in Nonlinear Wave Systems.- The Quantum Correlation Function as the ? Function of Classical Differential Equations.- Lattice Models in Statistical Mechanics and Soliton Equations.- Elementary Introduction to Quantum Groups.- Knot Theory and Integrable Systems.- Solitons and Computation.- Symplectic Aspects of Some Eigenvalue Algorithms.- Whiskered Tori for NLS Equations.- Index of Contributors.