Hamiltonian PDEs: new connections between dynamical systems and PDEs with small divisors phenomena

Many partial differential equations arising in physics can be seen as infinite dimensional Hamiltonian systems. Main examples are the nonlinear wave equation, the nonlinear Schrödinger equation, the beam, the membrane and the Kirkhoff equations in elasticity theory, the Euler equations of hydrodynamics as well as their approximate models like the KdV, the Camassa-Holm, the Boussinesq and the Kadomtsev-Petviashvili equations, De Gasperis-Procesi, etc. In the last years important mathematical progresses have been achieved in the study of these evolutionary Partial Differential Equations (PDEs) adopting the "dynamical systems philosophy". In particular, building on the experience gained from the qualitative study of finite dimensional dynamical systems, the search for periodic and quasi-periodic solutions was regarded as a first step toward better understanding also the complicated flow evolution of Hamiltonian PDEs. The analysis of the main structures of an infinite dimensional phase space such as periodic orbits, embedded invariant tori, center manifolds, etc., is an essential change of paradigm in the study of hyperbolic equations which has been recently very fruitful. The aim of this project is to pursue this program of research joining Dynamical Systems, KAM and Nash-Moser theory, Normal form theory, Variational Methods, Harmonic analysis and PDE tecnhiques, ...