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

We study spectral-Galerkin methods (SGM) for nonlinear eigenvalue problems, where the Legendre polynomials are used as the basis functions for the trial function space. The SGM is applied to find the ground state solution of the Gross–Pitaevskii equation (GPE) and the GPE in a periodic potential. When the SGM is incorporated in the context of continuation methods for curve-tracking, a manifest advantage is that the target points are very close to the bifurcation points. Thus it takes less continuation steps to reach the target point when compared with the centered difference methods (CDM) or finite element methods (FEM). We also implement the spectral collocation method (SCM) so that the computational cost for the numerical integration in the SGM can be saved. Comprehensive numerical experiments on the GPE using various numerical methods described in this paper are reported.