Reducibility and nonlinear stability for a quasi-periodically forced NLS

Motivated by the problem of long time stability vs. instability of KAM tori of the Nonlinear cubic Schr\"odinger equation (NLS) on the two dimensional torus $\mathbb T^2:= (\mathbb R/2\pi \mathbb Z)^2$, we consider a quasi-periodically forced NLS equation on $\mathbb T^2$ arising from the linearization of the NLS at a KAM torus. We prove a reducibility result as well as long time stability of the origin. The main novelty is to obtain the precise asymptotic expansion of the frequencies which allows us to impose Melnikov conditions at arbitrary order

A note on growth of Sobolev norms near quasiperiodic finite-gap tori for the 2D cubic NLS equation

We present the recent result in [29] concerning strong nonlinear instability and growth of Sobolev norms near quasiperiodic finite-gap solutions of the defocusing cubic nonlinear Schrodinger equation (NLS) on the two-dimensional torus. The equation admits a special family of elliptic invariant quasiperiodic tori called finite-gap solutions. These are inherited from the integrable 1D model (cubic NLS on the circle) by considering solutions that depend only on one variable. We construct solutions of the 2D cubic NLS that start arbitrarily close to such invariant tori in the H-s topology (0 < s < 1) and whose H-s norm can grow by any given factor

Growth of Sobolev norms for the quintic NLS on T2\mathbb T^2

We study the quintic Non Linear Schr\"odinger equation on a two dimensional torus and exhibit orbits whose Sobolev norms grow with time. The main point is to reduce to a sufficiently simple toy model, similar in many ways to the one used in the case of the cubic NLS. This requires an accurate combinatorial analysis.Comment: 41 pages, 5 figures. arXiv admin note: text overlap with arXiv:0808.1742 by other author

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