Approximation of a two-dimensional Gross–Pitaevskii equation with a periodic potential in the tight-binding limit

Pub Date : 2024-08-12 DOI:10.1002/mana.202300322
Steffen Gilg, Guido Schneider
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Abstract

The Gross–Pitaevskii (GP) equation is a model for the description of the dynamics of Bose–Einstein condensates. Here, we consider the GP equation in a two-dimensional setting with an external periodic potential in the x $x$ -direction and a harmonic oscillator potential in the y $y$ -direction in the so-called tight-binding limit. We prove error estimates which show that in this limit the original system can be approximated by a discrete nonlinear Schrödinger equation. The paper is a first attempt to generalize the results from [19] obtained in the one-dimensional setting to higher space dimensions and more general interaction potentials. Such a generalization is a non-trivial task due to the oscillations in the external periodic potential which become singular in the tight-binding limit and cause some irregularity of the solutions which are harder to handle in higher space dimensions. To overcome these difficulties, we work in anisotropic Sobolev spaces. Moreover, additional non-resonance conditions have to be satisfied in the two-dimensional case.

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二维格罗斯-皮塔耶夫斯基方程在紧束缚极限下与周期势的近似关系
格罗斯-皮塔耶夫斯基(GP)方程是描述玻色-爱因斯坦凝聚态动力学的模型。在这里,我们考虑的是在二维环境中的 GP 方程,在所谓的紧束缚极限中,在-方向上有一个外部周期势,在-方向上有一个谐振子势。我们证明的误差估计值表明,在此极限下,原始系统可以用离散非线性薛定谔方程来近似。本文首次尝试将 [19] 在一维环境下获得的结果推广到更高的空间维度和更一般的相互作用势。由于外部周期势的振荡在紧约束极限中变得奇异,并导致解的不规则性,这在更高的空间维度中更难处理,因此这种推广是一项非难事。为了克服这些困难,我们在各向异性的 Sobolev 空间中进行研究。此外,在二维情况下还必须满足额外的非共振条件。
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