Numerical solution to coupled nonlinear Schrödinger equations on unbounded domains

The numerical simulation of coupled nonlinear Schrödinger equations on unbounded domains is considered in this paper. By using the operator splitting technique, the original problem is decomposed into linear and nonlinear subproblems in a small time step. The linear subproblem turns out to be two decoupled linear Schrödinger equations on unbounded domains, where artificial boundaries are introduced to truncate the unbounded physical domains into finite ones. Local absorbing boundary conditions are imposed on the artificial boundaries. On the other hand, the coupled nonlinear subproblem is an ODE system, which can be solved exactly. To demonstrate the effectiveness of our method, some comparisons in terms of accuracy and computational cost are made between the PML approach and our method in numerical examples.     

Time-splitting pseudo-spectral domain decomposition method for the soliton solutions of the one- and multi-dimensional nonlinear Schrödinger equations

In this paper, we study the simulation of nonlinear Schrödinger equation in one, two and three dimensions. The proposed method is based on a time-splitting method that decomposes the original problem into two parts, a linear equation and a nonlinear equation. The linear equation in one dimension is approximated with the Chebyshev pseudo-spectral collocation method in space variable and the Crank–Nicolson method in time; while the nonlinear equation with constant coefficients can be solved exactly. As the goal of the present paper is to study the nonlinear Schrödinger equation in the large finite domain, we propose a domain decomposition method. In comparison with the single-domain, the multi-domain methods can produce a sparse differentiation matrix with fewer memory space and less computations. In this study, we choose an overlapping multi-domain scheme. By applying the alternating direction implicit technique, we extend this efficient method to solve the nonlinear Schrödinger equation both in two and three dimensions, while for the solution at each time step, it only needs to solve a sequence of linear partial differential equations in one dimension, respectively. Several examples for one- and multi-dimensional nonlinear Schrödinger equations are presented to demonstrate high accuracy and capability of the proposed method. Some numerical experiments are reported which show that this scheme preserves the conservation laws of charge and energy.     

Parallel Split-Step Fourier Methods for the Coupled Nonlinear Schrödinger Type Equations

The nonlinear Schrödinger type equations are of tremendous interest in both theory and applications. Various regimes of pulse propagation in optical fibers are modeled by some form of the nonlinear Schrödinger equation.In this paper we introduce parallel split-step Fourier methods for the numerical simulations of the coupled nonlinear Schrödinger equation that describes the propagation of two orthogonally polarized pulses in a monomode birefringent fibers. These methods are implemented on the Origin 2000 multiprocessor computer. Our numerical experiments have shown that these methods give accurate results and considerable speedup.     

Local artificial boundary conditions for Schrödinger and heat equations by using high-order azimuth derivatives on circular artificial boundary

The aim of the paper is to design high-order artificial boundary conditions for the Schrödinger equation on unbounded domains in parallel with a treatment of the heat equation. We first introduce a circular artificial boundary to divide the unbounded definition domain into a bounded computational domain and an unbounded exterior domain. On the exterior domain, the Laplace transformation in time and Fourier series in space are applied to achieve the relation of special functions. Then the rational functions are used to approximate the relation of the special functions. Applying the inverse Laplace transformation to a series of simple rational function, we finally obtain the corresponding high-order artificial boundary conditions, where a sequence of auxiliary variables are utilized to avoid the high-order derivatives in respect to time and space. Furthermore, the finite difference method is formulated to discretize the reduced initial–boundary value problem with high-order artificial boundary conditions on a bounded computational domain. Numerical experiments are presented to illustrate the performance of our method.     

High-order compact splitting multisymplectic method for the coupled nonlinear Schrödinger equations

In this paper, we develop a new kind of multisymplectic integrator for the coupled nonlinear Schrödinger (CNLS) equations. The CNLS equations are cast into multisymplectic formulation. Then it is split into a linear multisymplectic formulation and a nonlinear Hamiltonian system. The space of the linear subproblem is approximated by a high-order compact (HOC) method which is new in multisymplectic context. The nonlinear subproblem is integrated exactly. For splitting and approximation, we utilize an HOC–SMS integrator. Its stability and conservation laws are investigated in theory. Numerical results are presented to demonstrate the accuracy, conservation laws, and to simulate various solitons as well, for the HOC–SMS integrator. They are consistent with our theoretical analysis.     

A dissipative exponentially-fitted method for the numerical solution of the Schrödinger equation and related problems

In this paper a dissipative exponentially-fitted method for the numerical integration of the Schrödinger equation and related problems is developed. The method is called dissipative since is a nonsymmetric multistep method. An application to the the resonance problem of the radial Schrödinger equation and to other well known related problems indicates that the new method is more efficient than the corresponding classical dissipative method and other well known methods. Based on the new method and the method of Raptis and Cash a new variable-step method is obtained. The application of the new variable-step method to the coupled differential equations arising from the Schrödinger equation indicates the power of the new approach.     

A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients

We propose a compact split-step finite difference method to solve the nonlinear Schrödinger equations with constant and variable coefficients. This method improves the accuracy of split-step finite difference method by introducing a compact scheme for discretization of space variable while this improvement does not reduce the stability range and does not increase the computational cost. This method also preserves some conservation laws. Numerical tests are presented to confirm the theoretical results for the new numerical method by using the cubic nonlinear Schrödinger equation with constant and variable coefficients and Gross-Pitaevskii equation.     

Numerical solution of coupled nonlinear Schrödinger equation by Galerkin method

The coupled nonlinear Schrödinger equation models several interesting physical phenomena presents a model equation for optical fiber with linear birefringence. In this paper we derive a finite element scheme to solve this equation, we test this method for stability and accuracy, many numerical tests have been conducted. The scheme is quite accurate and describe the interaction picture clearly.     

A numerical Maxwell-Schrödinger model for intense laser-matter interaction and propagation

We present in this paper an original ab initio Maxwell-Schrödinger model and a methodology to simulate intense ultrashort laser pulses interacting with a 3D H⁺₂-gas in the nonlinear nonperturbative regime under and beyond Born-Oppenheimer approximation. The model we present is the first one to our knowledge (excepted in [E. Lorin, S. Chelkowski, A. Bandrauk, A Maxwell-Schrödinger model for non-perturbative laser-molecule interaction and some methods of numerical computation, Proceeding CRM, vol. 41, American Mathematics Society, 2007], where a one-dimensional version is presented) to be totally nonperturbative, vectorial and multidimensional, taking into account ionization, and high order nonlinearities going far beyond classical nonlinear Maxwell or Schrödinger models. After a presentation of the model and a short mathematical study, we examine some numerical approximations for its computation. In particular, we focus on the polarization computation allowing an efficient coupling between the Maxwell and time dependent Schrödinger equations (TDSE), and on an efficient parallelization. Examples of numerical computations of high order harmonic generation and of electric field propagation are presented for one molecule and up to 512, thus highlighting cooperative effects in harmonic generation at high order.     

Quasilinearization approach to computations with singular potentials

We pioneered the application of the quasilinearization method (QLM) to the numerical solution of the Schrödinger equation with singular potentials. The spiked harmonic oscillator r²+λr−α is chosen as the simplest example of such potential. The QLM has been suggested recently for solving the Schrödinger equation after conversion into the nonlinear Riccati form. In the quasilinearization approach the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of solutions near the boundaries.We show that the energies of bound state levels in the spiked harmonic oscillator potential which are notoriously difficult to compute for small couplings λ, are easily calculated with the help of QLM for any λ and α with accuracy of twenty significant figures.     

Soliton dynamics in linearly coupled discrete nonlinear Schrödinger equations

We study soliton dynamics in a system of two linearly coupled discrete nonlinear Schrödinger equations, which describe the dynamics of a two-component Bose gas, coupled by an electromagnetic field, and confined in a strong optical lattice. When the nonlinear coupling strengths are equal, we use a unitary transformation to remove the linear coupling terms, and show that the existing soliton solutions oscillate from one species to the other. When the nonlinear coupling strengths are different, the soliton dynamics is numerically investigated and the findings are compared to the results of an effective two-mode model. The case of two linearly coupled Ablowitz–Ladik equations is also briefly discussed.     

A semi-explicit multi-symplectic splitting scheme for a 3-coupled nonlinear Schrödinger equation

In this paper, we mainly propose an efficient semi-explicit multi-symplectic splitting scheme to solve a 3-coupled nonlinear Schrödinger (3-CNLS) equation. Based on its multi-symplectic formulation, the 3-CNLS equation can be split into one linear multi-symplectic subsystem and one nonlinear infinite-dimensional Hamiltonian subsystem. For the linear subsystem, the multi-symplectic Fourier pseudospectral method and symplectic Euler method are employed in spatial and temporal discretizations, respectively. For the nonlinear subsystem, the mid-point symplectic scheme is used. Numerical experiments for the unstable plane waves show the effectiveness of the proposed method during long-time numerical calculation.     

A Chebyshev pseudospectral multidomain method for the soliton solution of coupled nonlinear Schrödinger equations

The collision of solitary waves is an important problem in both physics and applied mathematics. In this paper, we study the solution of coupled nonlinear Schrödinger equations based on pseudospectral collocation method with domain decomposition algorithm for approximating the spatial variable. The problem is converted to a system of nonlinear ordinary differential equations which will be integrated in time by explicit Runge–Kutta method of order four. The multidomain scheme has much better stability properties than the single domain. Thus this permits using much larger step size for the time integration which fulfills stability restrictions. The proposed scheme reduces the effects of round-of-error for the Chebyshev collocation and also uses less memory without sacrificing the accuracy. The numerical experiments are presented which show the multidomain pseudospectral method has excellent long-time numerical behavior and preserves energy conservation property.     

A generator of dissipative methods for the numerical solution of the Schrödinger equation

In this paper a generator of hybrid methods with minimal phase-lag is developed for the numerical solution of the Schrödinger equation and related problems. The generator's methods are dissipative and are of eighth algebraic order. In order to have minimal phase-lag with the new methods, their coefficients are determined automatically. Numerical results obtained by their application to some well known problems with periodic or oscillating solutions and to the coupled differential equations of the Schrödinger type indicate the efficiency of these new methods.     

Interactions of two co-propagating laser beams in underdense plasmas using a generalized Peaceman-Rachford ADI form

A generalized Peaceman-Rachford ADI form based on the regularized finite difference scheme is employed to study the nonlinear interactions of two co-propagating laser beams in underdense plasmas. A numerical scheme using the form is constructed for the solution of two coupled 2D time-dependent nonlinear Schrödinger equations for quasineutral plasmas in paraxial approximation.In the constructed scheme, the coupled equations are reduced to systems of nonlinear tridiagonal equations that are solved using the matrix factorization and a direct explicit iteration method. In the stability analysis, a time-varying control parameter is introduced for a conditionally stable solution at varying conditions.The scheme has a good agreement with a previous simulations at similar conditions, and in comparison with other algorithms, the scheme presents a better conservation characteristic for the photons power. Finally, the simulations results confirmed that the scheme is suitable to simulate the interactions of two co-propagating laser beams in underdense plasma and it can successively simulate the associated phenomena at different parameters and conditions.     

Computing multiple peak solutions for Bose-Einstein condensates in optical lattices

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 second order finite difference approximations are used for computing energy levels of 3D NLS defined in a cubic box and a cylinder, respectively. We show that the choice of trapping potential plays a key role in computing energy levels of the NLS. We also investigate multiple peak solutions for BEC confined in optical lattices. Sample numerical results for the NLS defined in a cylinder and a cubic box are reported. Specifically, our numerical results show that the number of peaks for the ground state solutions of BEC in a periodic potential depends on the distance of neighbor wells.     

Symplectic and multi-symplectic methods for coupled nonlinear Schrödinger equations with periodic solutions

We consider for the integration of coupled nonlinear Schrödinger equations with periodic plane wave solutions a splitting method from the class of symplectic integrators and the multi-symplectic six-point scheme which is equivalent to the Preissman scheme. The numerical experiments show that both methods preserve very well the mass, energy and momentum in long-time evolution. The local errors in the energy are computed according to the discretizations in time and space for both methods. Due to its local nature, the multi-symplectic six-point scheme preserves the local invariants more accurately than the symplectic splitting method, but the global errors for conservation laws are almost the same.     

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.     

A family of high-order multistep methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation

Many simulation algorithms (chemical reaction systems, differential systems arising from the modelling of transient behaviour in the process industries etc.) contain the numerical solution of systems of differential equations. For the efficient solution of the above mentioned problems, linear multistep methods or Runge-Kutta single-step methods are used. For the simulation of chemical procedures the radial Schrödinger equation is used frequently. In the present paper we will study a class of linear multistep methods. More specifically, the purpose of this paper is to develop an efficient algorithm for the approximate solution of the radial Schrödinger equation and related problems. This algorithm belongs in the category of the multistep methods. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. Hence the main result of this paper is the development of an efficient multistep method for the numerical solution of systems of ordinary differential equations with oscillating or periodical solutions. The reason of their efficiency, as the analysis proved, is that the phase-lag and its derivatives are eliminated. Another reason of the efficiency of the new obtained methods is that they have high algebraic order     

Multi-symplectic splitting method for the coupled nonlinear Schrödinger equation

In this paper, we propose a multi-symplectic splitting method to solve the coupled nonlinear Schrödinger (CNLS) equation by using the idea of splitting the multi-symplectic partial differential equation (PDE). Numerical experiments show that the proposed method can simulate the propagation and collision of solitons well. The corresponding errors in global energy and momentum are also presented to show the good preservation property of the proposed method during long-time numerical calculation.