Numerical solution of the GrossPitaevskii equation for BoseEinstein condensation
We study the numerical solution of the time-dependent GrossPitaevskii equation (GPE) describing a BoseEinstein condensate (BEC) at zero or very low temperature. In preparation for the numerics we scale the 3d GrossPitaevskii 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 BoseEinstein condensation.
grosspitaevskii equation boseeinstein condensation (bec) time-splitting spectral method defocusing/focusing nonlinearity approximate ground state solution