I Give Up Study Guide: The Secret Joy of a Surrendered Life

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 Schrodinger 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 Frohlich 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.