KAM Theory for PDEs

Abstract

This thesis is concerned with KAM theory for Hamiltonian partial differential equations. This theory answers to the following matter: since the solutions of a linear equation are periodic, quasi periodic or almost periodic (for they are superpositions of periodic motions), the problem is to investigate what happens when we add a (sufficiently) small nonlinearity. This thesis contains two new results: an abstract KAM theorem for degenerate infinite--dimensional systems, with an application to the nonlinear wave equation, and a KAM theorem for a completely resonant nonlinear Schr\"odinger equation.