Numerical solution for the Gross-Pitaevskii equation

We solve the time-independent Gross-Pitaevskii (GP) equation which describes the dilute Bose-condensed atoms in harmonic trap at zero temperature by symplectic shooting method (SSM). Both the repulsive nonlinearity and the attractive nonlinearity cases are studied, and the bound state eigenvalues as well as the corresponding wavefunctions are evaluated. We also present the numerical results by studying the time-dependent GP equation, and comparisons are made between the results obtained by the time-independent approach and the time-dependent approach.