Analytical Solutions to the Time-Independent Gross-Pitaevskii Equation with a Harmonic Trap

We try to find the analytical solutions to the time-independent Gross-Pitaevskii equation, a nonlinear Schr?dinger equation used in the simulation of Bose–Einstein condensates trapped in a harmonic potential. Both the homotopy analysis method and the Galerkin spectral method are applied. We investigate the one-dimensional case and obtain the approximate analytical solutions successfully. Comparison between the analytical solutions and the numerical solutions has been made. The results indicate that they agree very well with each other when the atomic interaction is not too strong.