Premium
Asymptotic Behavior of the Nonlinear Schrödinger Equation with Harmonic Trapping
Author(s) -
Hani Zaher,
Thomann Laurent
Publication year - 2016
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21594
Subject(s) - trapping , scattering , nonlinear schrödinger equation , nonlinear system , mathematics , limit (mathematics) , dimension (graph theory) , harmonic , schrödinger equation , mathematical physics , mathematical analysis , zero (linguistics) , homogeneous , scattering theory , physics , quantum mechanics , pure mathematics , ecology , linguistics , philosophy , combinatorics , biology
We consider the cubic nonlinear Schrödinger equation with harmonic trapping on ℝ D (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 ℝ D −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.