Seminar

The study of solutions of Hamiltonian PDEs undergoing growth 

of Sobolev norms H^s (with s\neq 1) as time evolves has drawn 

considerable attention in recent years. The importance of growth of 

Sobolev norms is due to the fact that it implies that the solution 

transfers energy to higher modes.

Consider the defocusing cubic nonlinear Schr\"odinger equation (NLS) on 

the two-dimensional torus. The equation admits a special family of 

invariant quasiperiodic tori. These are inherited from the 1D cubic NLS 

(on the circle) by considering solutions that depend only on one 

variable. We show that, under certain assumptions, these tori are 

transversally unstable in Sobolev spaces $H^s$ ($0<s<1$). More 

precisely, we construct  solutions of the 2D cubic NLS which start 

arbitrarily close to such invariant tori in the $H^s$ topology and whose 

$H^s$ norm can grow by any given factor. This is a joint work with Z. 

Hani, E. Haus, A. Maspero and M. Procesi.