ERC- Starting Grants. Ideas. Hamiltonian PDEs and small divisors problems: a dynamical systems approach

A large number of partial differential equations of Physics have the structure of an infinite-dimensional Hamiltonian dynamical system. In this class of equations appear, among others, the Schrödinger equation (NLS), the wave equation (NLW), the Euler equations of hydrodynamics and the numerous models that derive from it. The study of these equations poses some fundamental questions that have inspired an entire research field in the last twenty years: the investigation of the main invariant structures of the phase space of a Hamiltonian system, starting from its stationary, periodic and quasi-periodic orbits. As in the case of finite-dimensional dynamical systems, one of the main problems in this field is linked to the well-known "small divisors problem''. A further difficulty is due to the fact that ''physically'' interesting equations, without outer parameters, are typically resonant and/or contain derivatives in the non-linearity. There are many fundamental open questions in this field. Our main goals are 1) the study of quasi--periodic solutions, in particular for semi-linear and quasi-linear equations 2) Study of normal forms, both in integrable and non--integrable cases. 3) Applications to hydrodynamics and search of quasi-periodic solutions in water wave problems. 4) Study of almost periodic solutions for semilinear PDEs. 5) quasi--periodic solutions for resonant systems (both finite and infinite dimensional) with minimal restrictions on the non--linearity.