Gibbs Measures and Quasi-Periodic Solutions for Nonlinear Hamiltonian PartialDifferential Equations

摘要

The purpose of this expose is to present a summary of some new developments inthe theory of Hamiltonian nonlinear evolution equations, more specifically, nonlinear Schrodinger equations. The themes and methods discussed are closely related to classical mechanics. The first topic is the existence of an invariant measure for the flow. This invariant measure is the (properly normalized) Gibbs measure from statistical mechanics and we establish wellposedness of the equation on its support. Results are obtained in 1D (in the focusing and defocusing case) and in the 2D defocusing case. The second topic concerns the persistency of time periodic and quasi-periodic solutions for Hamiltonian perturbations of linear and integrable equations. We follow a method, the so-called Liapounov-Schmidt decomposition. The main advantage of this technique is the fact that it overcomes certain limitations of the KAM scheme, which is necessary to deal in particular with the problems in space-dimension D = or > 2. This work is a new approach to KAM problems, also in finite dimensional phase space.