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

  • Authors:

  • Weizhu Bao;Dieter Jaksch;Peter A. Markowich

  • Affiliations:

  • Department of Computational Science, National University of Singapore, Singapore 117543, Singapore;Institut für Theoretische Physik, Universität Innsbruck, Innsbruck A-6020, Austria;Institute of Mathematics, University of Vienna, Boltzmanngasse 9, Vienna A-1090, Austria

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2003

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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 an 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.