Computing multiple peak solutions for Bose–Einstein condensates in optical lattices

We briefly review a class of nonlinear Schrodinger equations (NLS) which govern various physical phenomenon of Bose–Einstein condensation (BEC). We derive formulas for computing energy levels and wave functions of the Schrodinger equation defined in a cylinder without interaction between particles. Both fourth order and second order finite difference approximations are used for computing energy levels of 3D NLS defined in a cubic box and a cylinder, respectively. We show that the choice of trapping potential plays a key role in computing energy levels of the NLS. We also investigate multiple peak solutions for BEC confined in optical lattices. Sample numerical results for the NLS defined in a cylinder and a cubic box are reported. Specifically, our numerical results show that the number of peaks for the ground state solutions of BEC in a periodic potential depends on the distance of neighbor wells.