Numerical Continuation for Nonlinear SchrÖdinger Equations

We discuss numerical methods for studying numerical solutions of N-coupled nonlinear Schrödinger equations (NCNLS), N = 2, 3. First, we discretize the equations by centered difference approximations. The chemical potentials and the coupling coefficient are treated as continuation parameters. We show how the predictor–corrector continuation method can be exploited to trace solution curves and surfaces of the NCNLS, where the preconditioned Lanczos method with iterative refinement is used as the linear solver. When the chemical potential is large enough, we obtain peak solutions of the NCNLS for certain values of the coupling coefficient. The contours of the peak solutions resemble those of the experimental results of Anglin and Ketterle [2002], and Anderson et al. [1995].