Many problems in nonlinear PDE which are of physical significance can be posed as Hamiltonian systems: Some principal examples include the nonlinear wave equations, the nonlinear Schrödinger equation, the KdV equation and the Euler equations of fluid mechanics. Complementing the theory of the initial value problem, it is natural to pose the question of stability of solutions for all times, and to describe the principal structures of phase space which are invariant under the flow. The subject of this volume is the development of extensions of KAM theory of invariant tori for PDE, for which the phase space is naturally infinite dimensional. The book starts with the definition of a Hamiltonian system in infinite dimensions. It reviews the classical theory of periodic solutions for finite dimensional dynamical systems, commenting on the role played by resonances. It then develops a direct approach to KAM theory in infinite dimensional settings, applying it to several of the PDE of interest. The volume includes a description of the methods of Fröhlich and Spencer for resolvant expansions of linear operators, as it is a basic technique used in this approach to KAM theory. The final chapter gives a presentation of the more recent developments of the subject. Text is in French.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

Readership

Graduate students and research mathematicians.

Table of Contents

• Introduction
• Équations aux dérivées partielles
• Solutions périodiques: le théorème du centre de Lyapunov et la théorie de A. Weinstein et J. Moser
• La théorie KAM en dimension infinie: énoncé des résultats
• La méthode de Nash-Moser
• Analyse du problème linéarisé
• Approche directe de la théorie KAM
• La théorie KAM en dimension infinie: autres résultats récents
• Bibliographie