Research

My research interests concern the analysis of nonlinear evolution PDE's on compact manifolds, in particular the techniques and ideas coming from the dynamical systems theory, such as perturbative methods. In particular, I am interested in the existence of Cantor manifolds of periodic and quasi-periodic solutions, long-time behaviour of solutions with initial data close to the equilibrium and stability or instability issues for Hamiltonian PDE's. Actually, many notable equations, like the nonlinear Schr\"odinger, the Korteweg de Vries, the Klein-Gordon and many others equations possess a Hamiltonian structure, which plays a fundamental role especially to obtain KAM and stability results. These systems, around an elliptic fixed point, behave like a (infinite) chain of harmonic oscillators coupled by the nonlinearities and a dynamical system approach often allows to make a qualitative portrait of the solutions in a neighborhood of the equilibrium.

In the last years I focused my research on quasi-linear Hamiltonian PDE's, namely equations such that the linear and the nonlinear terms contain derivatives of the same order. These kind of equations often arise in physical applications, especially in fluid dynamics problems. In the "quasi-linear context" the progresses on the search for quasi-periodic solutions, extensions of Birkhoff normal form methods and reducibility results are very recent.

The main issue in these problems is related to serious small divisors difficulties, which arise from the presence of resonances.

In the infinite-dimensional framework, these phenomena complicate the dynamics also in a small neighborhood of the equilibrium.

During my Ph.D experience I worked on KAM problems related to quasi-linear Hamiltonian PDEs which come from shallow water models. In particular I studied the existence and stability of small amplitude quasi-periodic solutions for generalized quasi-linear KdV-type equations and Hamiltonian perturbations of the Degasperis-Procesi equation.

These problems involve Birkhoff normal form methods, Pseudo differential calculus and the Nash-Moser theory.