Infinite energy cascades and modified scattering for the cubic Schr\"odinger on product spaces

PDE Seminar
Thursday, April 17, 2014 - 15:05
1 hour (actually 50 minutes)
Skiles 006
New York University
We consider the cubic nonlinear Schr\"odinger equation posed on the product spaces \R\times \T^d. We prove the existence of global solutions exhibiting infinite growth of high Sobolev norms. This is a manifestation of the "direct energy cascade" phenomenon, in which the energy of the system escapes from low frequency concentration zones to arbitrarily high frequency ones (small scales). One main ingredient in the proof is a precise description of the asymptotic dynamics of the cubic NLS equation when 1\leq d \leq 4. More precisely, we prove modified scattering to the resonant dynamics in the following sense: Solutions to the cubic NLS equation converge (as time goes to infinity) to solutions of the corresponding resonant system (aka first Birkhoff normal form). This is joint work with Benoit Pausader (Princeton), Nikolay Tzvetkov (Cergy-Pontoise), and Nicola Visciglia (Pisa).