Asymptotic behavior of the nonlinear Schrödinger equation with harmonic trapping

Abstract

We consider the cubic nonlinear Schrödinger equation with harmonic trapping on ${\mathbb{R}}^D$ ($1\le D\le 5$). In the case when all but one directions are trapped (a.k.a "cigar-shaped" trap), following the approach of Hani-Pausader-Tzvetkov-Visciglia, we prove modified scattering and construct modified wave operators for small initial and final data respectively. The asymptotic behavior turns out to be a rather vigorous departure from linear scattering and is dictated by the resonant system of the NLS equation with full trapping on ${\mathbb{R}}^{D-1}$. In the physical dimension $D=3$, this system turns out to be exactly the (CR) equation derived and studied by Faou-Germain-Hani. The special dynamics of the latter equation, combined with the above modified scattering results, allow to justify and extend some physical approximations in the theory of Bose-Einstein condensates in cigar-shaped traps.

40 pages