Degenerate KAM Theory for PDEs

This thesis deals with KAM theory for Hamiltonian partial differential equations. This theory concerns the problem of the persistence of normal mode oscillations (periodic, quasi periodic or almost periodic solutions) of a linear PDE 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 Schrödinger equation.