2014/08/26 by Zaher Hani, Laurent Thomann, Hani, Zaher +1 · 1 citation
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AP #math.MP
paper · pdf · doi:10.48550/arxiv.1408.6213
40 pages
arxiv created 2014/08/26 · arxiv updated 2014/08/27
We consider the cubic nonlinear Schrödinger equation with harmonic trapping on ℝD (1≤ D≤ 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 ℝD-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.