KAM for quasi-linear PDE's

In this Thesis we present two new results of existence and stability of Cantor families of small amplitude quasi-periodic in time solutions for quasi-linear Hamiltonian PDE's arising as models for shallow water phenomena.\\ The considered problems present serious small divisors difficulties and the results are achieved by implementing Nash-Moser algorithms and by exploiting pseudo differential calculus techniques. \smallskip The first result concerns a generalized quasi-linear KdV equation \[ u_t+u_{xxx}+\mathcal{N}_2(x, u, u_x, u_{xx}, u_{xxx})=0, \quad x\in \T, \] where $\mathcal{N}_2$ is a nonlinearity originating from a cubic Hamiltonian.\\ The nonlinear part depends upon some parameters and it is intriguing to study how the choice of these parameters affects the bifurcation analysis.\\ The linearized equation at the origin is resonant, namely the linear solutions are all periodic, hence the existence of the expected quasi-periodic solutions is due only to the presence of the nonlinearities.\\ The nonlinear terms of these equations are quadratic and contains derivatives of the same order of the linear part, thus they produce strong perturbative effect near the origin. \smallskip The second result is the first KAM result for quasi-linear PDE's with asymptotically linear dispersion law and it implies the first existence result for quasi-periodic solutions of the Degasperis-Procesi equation.\\ We consider Hamiltonian perturbations of the Degasperis-Procesi equation \[ u_t-u_{x x t}+u_{xxx}-4 u_x-u u_{xxx}-3 u_x u_{xx}+4 u u_x+\mathcal{N}_6( u, u_x, u_{xx}, u_{xxx})=0, \quad x\in \T, \] where $\mathcal{N}_6$ is a nonlinearity originating from a Hamiltonian density with a zero of order seven at the origin.\\ We exploit the integrable structure of the unperturbed equation $\mathcal{N}_6=0$ to overcome some small divisors problems.\\ The complicated symplectic structure and the asymptotically linear dispersion law make harder the analysis of the linearized operator in a neighborhood of the origin, which is required by the Nash-Moser scheme, and the measure estimates for the frequencies of the expected quasi-periodic solutions