Adaptive continuation algorithms for computing energy levels of rotating Bose–Einstein condensates

We describe adaptive continuation algorithms for computing energy levels of the Bose–Einstein condensates (BEC) with emphasis on the rotating BEC. We show that the rotating BEC in the complex plane is governed by special two-coupled nonlinear Schrödinger equations (NLS) in the real domain, which makes the eigenvalues of the discrete coefficient matrix at least double. A predictor–corrector continuation method is used to trace solution curves of the rotating BEC defined in square domains. The wave functions of the rotating BEC can be easily obtained whenever the solution curves of the two-coupled NLS are numerically traced. From the physical point of view, the proposed algorithm has the advantage that the energy levels of the system are computed intuitively, where the energy information of the associated Schrödinger eigenvalue problem is fully exploited. The superfluid density we obtain on the first solution branch resembles the figure shown in [J.R. Anglin, W. Ketterle, Nature 416 (2002) 211]. We also obtain superfluid densities on the other solution branches, which to the best of our knowledge, have never shown up in the literatures.