PRIN 2009: Critical point theory and perturbative methods for nonlinear differential equations. Hamiltonian Systems and Partial Differential Equations

Many partial differential equations arising in Physics can be seen as innite dimensional Hamiltonian systems. Main examples are the nonlinear Schrodinger (NLS) and wave (NLW) equations, the beam, membrane and Kirkhoff equations in elasticity theory, the Euler equations of hydrodynamics as well as their approximate models like KdV, 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 and using tools from “Critical Point Theory”. The analysis of the principal structures of an infinite dimensional phase space such as stationary solutions, periodic orbits, solitons, embedded invariant tori, center manifolds, as well as stable and unstable manifolds, is an essential change of paradigm in the study of PDEs. The plan is to pursue this fascinating research, in particular, for 1. PDEs (like NLS and NLW) in any spatial dimension. 2. Quasi-linear PDEs with nonlinearities which depend on derivatives, like the Euler equations of hydrodynamics, and their approximate models. 3. Almost periodic solutions. 4. Stationary solutions for PDE's with nonlocal terms 5. Point-particle limit evolution of solitons for NLS 6. Qualitative properties and asymptotic behaviour of solutions of Allen-Cahn type PDE’s