Aims and scope:

A huge variety of physical systems is modeled by Hamiltonian dispersive Partial Differential Equations (PDEs), as the nonlinear Schrödinger (NLS) and wave (NLW) equations, the Euler and the water waves equations, KdV, etc... 

In the last years, the study of Hamiltonian and Dispersive PDEs has seen outstanding advances obtained by combining a dynamical systems approach, based on normal forms, KAM theory and chaotic dynamics techniques, with dispersive and energy estimates for the evolutionary flow, together with probabilistic arguments.

The purpose of this project is to gather leading Italian mathematicians working on Hamiltonian systems and Dispersive PDEs under complementary perspectives to address fundamental open problems regarding:

A. KAM for PDEs

B. Normal form and long time existence results

C. Unstable dynamics

D. Probabilistic dispersive PDEs

E. Asymptotic behavior of linear and nonlinear flows.

More specifically, the project addresses the following challenging themes:

A. KAM for PDEs: Quasi-periodic solutions in fluid equations, like vortex patches, Water Waves and Navier-Stokes equations, as well as for NLS on higher dimensional manifolds. Almost periodic solutions with Sobolev regularity for NLS. Periodic solutions for the Vlasov equation and periodic solitons of NLS.

B. Normal forms: Upper bounds for the growth of the Sobolev norm for unbounded perturbations of NLS on T^d or other higher dimensional manifolds, as well as long time existence results for water waves and Kirchhoff equations.

C. Unstable dynamics: Lower bounds for the growth of Sobolev norms, construction of special unstable trajectories close to KAM solutions of NLS on T^2, and also irrational tori. For non gauge invariant NLS construction of blow-up solutions. Prove Benjamin-Feir instability results for space periodic traveling water waves under long-wave periodic perturbations.

D. Probabilistic Dispersive PDEs: Dispersive equations can be fruitfully investigated either from a deterministic and a probabilistic point of view. For probabilistic PDEs, relevant open problems concern the evolution of Gaussian measures along the deterministic flow of NLS and gKdV as well as NLS with multiplicative spatial white noise. We also plan to construct invariant measures for defocusing equations in noncompact situations.

E. Asymptotic behavior of flows: An essential tool in all investigations of nonlinear dispersive equations is the knowledge of the asymptotic properties and structure of dispersive flows. Crucial open problems in this area concern linear flows perturbed by critical and scale invariant potentials, which are physically relevant; flows generated by non selfadjoint Hamiltonians; and flows on manifolds with large curvature at infinity. We are also interested on stability issues of solitons of NLS via the nonlinear Fermi Golden Rule.