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arXiv · 1408.6213

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

Zaher Hani·Laurent Thomann
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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.

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Zaher Hani, Laurent Thomann. 2014-08-26. Asymptotic behavior of the nonlinear Schr\"odinger equation with harmonic trapping. https://arxiv.org/abs/1408.6213

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