- Series
- PDE Seminar
- Time
- Thursday, April 17, 2014 - 3:05pm for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Zaher Hani – New York University
- Organizer
- Geng Chen
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).