Asymptotic behavior of the nonlinear Schr\"odinger equation with harmonic trapping

TL;DR

This paper investigates the long-term behavior of the cubic nonlinear Schrödinger equation with harmonic trapping in dimensions 1 to 5, revealing modified scattering phenomena influenced by resonant systems, with implications for Bose-Einstein condensates.

Contribution

It proves modified scattering and constructs wave operators for the NLS with partial trapping, extending understanding of asymptotic dynamics in physically relevant trapping geometries.

Findings

Modified scattering established for small data in cigar-shaped traps.

Asymptotic behavior governed by resonant systems related to full trapping.

Results support physical models of Bose-Einstein condensates in elongated traps.

Abstract

We consider the cubic nonlinear Schr\"odinger equation with harmonic trapping on $\mathbb{R}^D$ ($1\leq D\leq 5$). 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.