A spectral-Galerkin continuation method using Chebyshev polynomials for the numerical solutions of the Gross–Pitaevskii equation

We study an efficient spectral-Galerkin continuation method (SGCM) and two-grid centered difference approximations for the numerical solutions of the Gross–Pitaevskii equation (GPE), where the second kind Chebyshev polynomials are used as the basis functions for the trial function space. Some basic formulae for the SGCM are derived so that the eigenvalues of the associated linear eigenvalue problems can be easily computed. The SGCM is implemented to investigate the ground and first excited-state solutions of the GPE. Both the parabolic and quadruple-well trapping potentials are considered. We also study Bose–Einstein condensates (BEC) in optical lattices, where the periodic potential described by the sine or cosine functions is imposed on the GPE. Of particular interest here is the investigation of symmetry-breaking solutions. Sample numerical results are reported.