Analysis Seminar

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Asymptotic Behavior for the Nonlinear Schrödinger Equation with Partially Periodic Data

Speaker: Benoit Pausader, Princeton University

Location: Warren Weaver Hall 1302

Date: Wednesday, December 4, 2013, 11 a.m.

Synopsis:

We consider the NLS equation on quotients of R^d, focusing on the case of RxT2. The question is to explore the asymptotic behavior of solutions in a more “compact” setting. We show how the scattering theory in the quintic case (the equivalent of the mass-critical case) is affected by the “smaller” volume and how, in the cubic case the asymptotic behavior is strongly modified by the presence of a secondary dynamics in logarithmic time. In the case of R or RxT (completely integrable case), this secondary dynamics can be explicitely integrated and only causes a phase correction. In the case of RxT^d, d>=2, this dynamics is more complicated and leads to new regimes. In particular, in this case, one can find global solutions which start arbitrarily small in H^s and grow unboundedly with time.

This is a joint work with Z. Hani as well as (for the cubic case) N. Tzvetkov and N. Visciglia.