ABSTRACT:

Our aim is to explain a normal form technic that allow to study the long time behaviour of the solutions of Hamiltonian perturbations of integrable systems. Our approach is centered on a Birkhoff normal form theorem in infinite dimension. We will apply this theorem to obtain dynamical results for some Hamiltonian PDEs including the NLS and NLW equations in one space dimension and for Dirichlet or periodic boundary conditions. Our method allows to prove the almost global existence of solutions for the Klein Gordon equation on a sphere (of any dimension) with small initial Cauchy data. This is a work in collaboration with D. Bambusi.