Continuation and stability analysis for Bloch waves of the Gross-Pitaevskii equation

We describe a two-parameter continuation algorithm for computing Bloch waves of Bose-Einstein condensates (BEC) in optical lattices which is governed by the Gross-Pitaevskii equation (GPE). The Fourier collocation method and fourth-order Adini’s elements with penalty are used to discretize the GPE. We propose two different approaches so that the two-parameter continuation algorithm can be modified to compute closed tubes at the four corners of the Bloch band. We also study linear stability analysis for the GPE. We show that all the discrete steady-state solutions are numerically neutrally stable. Numerical results show that the four edges of the Bloch waves are surrounded by closed loops if the coefficient of the cubic nonlinear term is greater than that of the periodic potential. Moreover, closed tubes at the four corners of the Bloch band are obtained. The numerical results display superfluidity of BEC.