Long-time Behavior of the Nonlinear Schrodinger Equation with Harmonic Trapping
Friday, November 4, 2016 - 9:00am - 9:50am
We consider the nonlinear Schrodinger equation either with full or partial harmonic trapping. In both cases, the long-time behavior is heavily influenced by the resonant part of the dynamics, which we shall define and study. In the case, when all directions but one are trapped (“cigar-shaped” trapping), we prove modified scattering of the nonlinear solutions towards the solutions of an appropriate resonant system. The results are particularly interesting in dimension D=3. There, this resonant system turns out to be essentially the same as the continuous resonant (CR) equation derived by Faou-Germain-Hani. The special dynamics of the latter equation seem to give insight into the dynamics of vortices in the theory of Bose-Einstein condensation. This is joint work with Laurent Thomann.