Computing wave functions of nonlinear Schrödinger equations: A time-independent approach

Abstract

We present a novel algorithm for computing the ground-state and excited-state solutions of M-coupled nonlinear Schrödinger equations (MCNLS). First we transform the MCNLS to the stationary state ones by using separation of variables. The energy level of a quantum particle governed by the Schrödinger eigenvalue problem (SEP) is used as an initial guess to computing their counterpart of a nonlinear Schrödinger equation (NLS). We discretize the system via centered difference approximations. A predictor–corrector continuation method is exploited as an iterative method to trace solution curves and surfaces of the MCNLS, where the chemical potentials are treated as continuation parameters. The wave functions can be easily obtained whenever the solution manifolds are numerically traced. The proposed algorithm has the advantage that it is unnecessary to discretize or integrate the partial derivatives of wave functions. Moreover, the wave functions can be computed for any time scale. Numerical results on the ground-state and excited-state solutions are reported, where the physical properties of the system such as isotropic and nonisotropic trapping potentials, mass conservation constraints, and strong and weak repulsive interactions are considered in our numerical experiments.