Dynamics of Hamiltonian PDEs

In this talk I will discuss some results about long time behaviors of solutions to Hamiltonian PDEs (Schrödinger, Quantum Harmonic Oscillator and Schrödinger-Poisson). In particular I will focus on a recent result where we (with J. Bernier and B. Grébert) prove exponential stability of small typical solutions of Schrödinger-Poisson equation by the so-called Rational Normal Form. For these resonant Hamiltonian PDEs the linear frequencies are fully resonant and we have to use the nonlinearity to break the resonances, which leads to a kind of new small divisors compared to Birkhoff Normal Form.