Growth of Sobolev norms for the cubic NLS near 1D quasi-periodic solutions

Abstract: Consider the defocusing cubic Schrödinger equation defined in the 2 dimensional torus. It has as a subsystem the one dimension cubic NLS (just considering solutions depending on one variable). The 1D equation is integrable and admits global action angle coordinates. Therefore, all its solutions are either periodic, quasi-periodic or almost-periodic. Consider one of the finite dimensional quasiperiodic invariant tori that the 1D equation possesses. Under certain assumptions on the torus (smallness, Diophantine frequency), we show that there exist solutions of the 2D equation which start arbitrarily close to this invariant torus in the H^s topology (with 0<s<1) and whose H^s Sobolev norm can grow by any given factor. This is a joint work with Z. Hani, E. Haus, A. Maspero and M. Procesi.