Numerical Solution of the Gross-Pitaevskii Equation for Bose-Einstein Condensation

TL;DR

This paper develops and analyzes a numerical method based on spectral techniques for solving the time-dependent Gross-Pitaevskii equation, enabling detailed simulations of Bose-Einstein condensates across various regimes.

Contribution

It introduces a scaled four-parameter model of the GPE and applies a time-splitting spectral method, supported by perturbation theory, for efficient numerical solutions in multiple dimensions.

Findings

Effective numerical scheme for 1d, 2d, and 3d GPEs

Demonstrates accuracy across weak/strong interactions

Provides insights into BEC physics through simulations

Abstract

We study the numerical solution of the time-dependent Gross-Pitaevskii equation (GPE) describing a Bose-Einstein condensate (BEC) at zero or very low temperature. In preparation for the numerics we scale the 3d Gross-Pitaevskii equation and obtain a four-parameter model. Identifying `extreme parameter regimes', the model is accessible to analytical perturbation theory, which justifies formal procedures well known in the physical literature: reduction to 2d and 1d GPEs, approximation of ground state solutions of the GPE and geometrical optics approximations. Then we use a time-splitting spectral method to discretize the time-dependent GPE. Again, perturbation theory is used to understand the discretization scheme and to choose the spatial/temporal grid in dependence of the perturbation parameter. Extensive numerical examples in 1d, 2d and 3d for weak/strong interactions, defocusing/focusing nonlinearity, and zero/nonzero initial phase data are presented to demonstrate the power of the numerical method and to discuss the physics of Bose-Einstein condensation.