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We present a novel algorithm for computing the ground-state and excited-state solutions of M-coupled nonlinear Schrö dinger 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 Schrö dinger eigenvalue problem (SEP) is used as an initial guess to… (More)

For most compute environments, adaptive forward differencing is much more efficient when performed using integer arithmetic than when using floating point. Previously low precision integer methods suffered from serious precision problems due to the error accumulation inherent to forward differencing techniques. This paper proposes several different… (More)

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… (More)

SUMMARY We study the Lanczos method for solving symmetric linear systems with multiple right-hand sides. First, we propose a numerical method of implementing the Lanczos method, which can provide all approximations to the solution vectors of the remaining linear systems. We also seek possible application of this algorithm for solving the linear systems that… (More)

We describe a special Gauss-Newton method for tracing solution manifolds with singularities of multi-parameter systems. First we choose one of the parameters as the continuation parameter, and fix the others. Then we trace one-dimensional solution curves by using continuation methods. Singularities such as folds, simple and multiple bifurcations on each… (More)

We study numerical methods for solving nonlinear elliptic eigenvalue problems which contain folds and bifurcation points. First we present some convergence theory for the MINRES, a variant of the Lanczos method. A multigrid-Lanczos method is then proposed for tracking solution branches of associated discrete problems and detecting singular points along… (More)

We study a finite difference continuation (FDC) method for computing energy levels and wave functions of Bose–Einstein condensates (BEC), which is governed by the Gross–Pitaevskii equation (GPE). We choose the chemical potential λ as the continuation parameter so that the proposed algorithm can compute all energy levels of the discrete GPE. The GPE is… (More)

We discuss numerical methods for studying numerical solutions of N-coupled nonlinear Schrödin-ger 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… (More)

a r t i c l e i n f o a b s t r a c t We briefly review a class of nonlinear Schrödinger equations (NLS) which govern various physical phenomenon of Bose–Einstein condensation (BEC). We derive formulas for computing energy levels and wave functions of the Schrödinger equation defined in a cylinder without interaction between particles. Both fourth order and… (More)