Numerical solution of the Gross–Pitaevskii equation for Bose–Einstein condensation

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.