|Location:||MSRI: Simons Auditorium|
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.