Direct and Inverse Methods in Nonlinear Evolution Equations: Lectures Given at the C.I.M.E. Summer School Held in Cetraro, Italy, September 5-12, 1999

Robert Conte3540200878, 9783540200871

Many physical phenomena are described by nonlinear evolution equation. Those that are integrable provide various mathematical methods, presented by experts in this tutorial book, to find special analytic solutions to both integrable and partially integrable equations. The direct method to build solutions includes the analysis of singularities � la Painlevé, Lie symmetries leaving the equation invariant, extension of the Hirota method, construction of the nonlinear superposition formula. The main inverse method described here relies on the bi-hamiltonian structure of integrable equations. The book also presents some extension to equations with discrete independent and dependent variables. The different chapters face from different points of view the theory of exact solutions and of the complete integrability of nonlinear evolution equations. Several examples and applications to concrete problems allow the reader to experience directly the power of the different machineries involved.

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Table of contents :

Content:
Exact solutions of nonlinear partial differential equations by singularity analysis….Pages 1-83
The method of Poisson pairs in the theory of nonlinear PDEs….Pages 85-136
Nonlinear superposition formulae of integrable partial differential equations by the singular manifold method….Pages 137-170
Hirota bilinear method for nonlinear evolution equations….Pages 171-222
Lie groups, singularities and solutions of nonlinear partial differential equations….Pages 223-273