Physics – Condensed Matter
Scientific paper
2003-03-13
J. Comput. Phys., Vol. 187, No. 1, pp. 318 - 342, 2003.
Physics
Condensed Matter
33 pages, 11 figures. Journal of Computational Physics, to appear
Scientific paper
10.1016/S0021-9991(03)00102-5
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.
Bao Weizhu
Jaksch Dieter
Markowich Peter A.
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