| ¡@¡@In this talks, we present a numerical method to compute the ground state solution of a Bose-Einstein condensate¡]BEC¡^by minimizing a functional and study the numerical solution of the time-dependent Gross-Pitaevskii equation¡]GPE¡^describing a BEC at zero or very low temperature. We take the 3d Gross-Pitaevskii equation, and as preparatory steps scale it to obtain a four-parameter model and use an approach well known in the physical literature to reduce it to 2d and 1d GPEs in certain limiting regimes. We provide approximate ground state solutions of GPE in two extreme regimes¡G¡]very¡^weak interactions and strong repulsive interaction. Then we use a time-splitting spectral method to solve the time-depend GPE in 1d, 2d, and 3d. Extensive numerical examples in 1d, 2d, and 3d for weak/strong interactions, defocusing/focusing nonlinearity, and zero/non zero initial phase data are presented to demonstrate the power of the numerical methods. |