Asymptotic Behavior of the Nonlinear Schrödinger Equation with Harmonic Trapping

Abstract

AbstractWe consider the cubic nonlinear Schrödinger equation with harmonic trapping on ℝD (1 ≤ D ≤ 5). In the case when all directions but one are trapped (aka “cigar‐shaped trap”), 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 by Faou, Germain, and the first author as the large box limit of the resonant NLS equation in the homogeneous (zero potential) setting. The special dynamics of the latter equation, combined with the above modified scattering results, allow us to justify and extend some physical approximations in the theory of Bose‐Einstein condensates in cigar‐shaped traps.© 2016 Wiley Periodicals, Inc.