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

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 $x$-direction and a harmonic oscillator potential in the $y$-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.