Mathematics

First, I will recall the classical theory of Birkhoff normal forms for partial differentials equations. This theory provides a sufficient condition of non-resonance to ensure that the dynamics of a perturbed integrable Hamiltonian system are similar, for very long times, to the dynamics of the unperturbed system. Then I will discuss two extensions of this construction when the non-resonance condition is no longer satisfied. We will deal with the examples of some resonant nonlinear Schrödinger equations on the one dimensional torus and the nonlinear Klein-Gordon equations on the d-dimensional torus with $d\geq 2$. This talk will rely on some joint works with Erwan Faou and Benoît Grébert.