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.     

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.     

High-order compact ADI (HOC-ADI) method for solving unsteady 2D Schrödinger equation

In this paper, a high-order compact (HOC) alternating direction implicit (ADI) method is proposed for the solution of the unsteady two-dimensional Schrödinger equation. The present method uses the fourth-order Padé compact difference approximation for the spatial discretization and the Crank-Nicolson scheme for the temporal discretization. The proposed HOC-ADI method has fourth-order accuracy in space and second-order accuracy in time. The resulting scheme in each ADI computation step corresponds to a tridiagonal system which can be solved by using the one-dimensional tridiagonal algorithm with a considerable saving in computing time. Numerical experiments are conducted to demonstrate its efficiency and accuracy and to compare it with analytic solutions and numerical results established by some other methods in the literature. The results show that the present HOC-ADI scheme gives highly accurate results with much better computational efficiency.     

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.     

Multi-symplectic wavelet collocation method for the nonlinear Schrödinger equation and the Camassa–Holm equation

In this paper, we develop a novel multi-symplectic wavelet collocation method for solving multi-symplectic Hamiltonian system with periodic boundary conditions. Based on the autocorrelation function of Daubechies scaling functions, collocation method is conducted for the spatial discretization. The obtained semi-discrete system is proved to have semi-discrete multi-symplectic conservation laws and semi-discrete energy conservation laws. Then, appropriate symplectic scheme is applied for time integration, which leads to full-discrete multi-symplectic conservation laws. Numerical experiments for the nonlinear Schrödinger equation and Camassa–Holm equation show the high accuracy, effectiveness and good conservation properties of the proposed method.     

Solitary wave solutions for a higher order nonlinear Schrödinger equation

We consider a higher order nonlinear Schrödinger equation with third- and fourth-order dispersions, cubic–quintic nonlinearities, self steepening, and self-frequency shift effects. This model governs the propagation of femtosecond light pulses in optical fibers. In this paper, we investigate general analytic solitary wave solutions and derive explicit bright and dark solitons for the considered model. The derived analytical dark and bright wave solutions are expressed in terms of the model coefficients. These exact solutions are useful to understand the mechanism of the complicated nonlinear physical phenomena which are related to wave propagation in a higher-order nonlinear and dispersive Schrödinger system.     

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.     

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.     

Pentadiagonal alternating-direction-implicit finite-difference time-domain method for two-dimensional Schrödinger equation

In this paper, we have proposed a pentadiagonal alternating-direction-implicit (Penta-ADI) finite-difference time-domain (FDTD) method for the two-dimensional Schrödinger equation. Through the separation of complex wave function into real and imaginary parts, a pentadiagonal system of equations for the ADI method is obtained, which results in our Penta-ADI method. The Penta-ADI method is further simplified into pentadiagonal fundamental ADI (Penta-FADI) method, which has matrix-operator-free right-hand-sides (RHS), leading to the simplest and most concise update equations. As the Penta-FADI method involves five stencils in the left-hand-sides (LHS) of the pentadiagonal update equations, special treatments that are required for the implementation of the Dirichlet’s boundary conditions will be discussed. Using the Penta-FADI method, a significantly higher efficiency gain can be achieved over the conventional Tri-ADI method, which involves a tridiagonal system of equations.     

CP methods of higher order for Sturm-Liouville and Schrödinger equations

The algorithm upon which the code SLCPM12, described in Computer Physics Communications 118 (1999) 259-277, is based, is extended to higher order. The implementation of the original algorithm, which was of order {12,10} (meaning order 12 at low energies and order 10 at high energies), was more efficient than the well-established codes SL02F, SLEDGE and SLEIGN. In the new algorithm the orders {14,12}, {16,14} and {18,16} are introduced. Besides regular Sturm-Liouville and one-dimensional Schrödinger problems also radial Schrödinger equations are considered with potentials of the form V(r)=S(r)/r+R(r), where S(r) and R(r) are well behaved functions which tend to some (not necessarily equal) constants when r→0 and r→∞. Numerical illustrations are given showing the accuracy, the robustness and the CPU-time gain of the proposed algorithms.     

An algebraic method for Schrödinger equations in quaternionic quantum mechanics

In the study of theory and numerical computations of quaternionic quantum mechanics and quantum chemistry, one of the most important tasks is to solve the Schrödinger equation with A an anti-self-adjoint real quaternion matrix, and |f〉 an eigenstate to A. The quaternionic Schrödinger equation plays an important role in quaternionic quantum mechanics, and it is known that the study of the quaternionic Schrödinger equation is reduced to the study of quaternionic eigen-equation Aα=αλ with A an anti-self-adjoint real quaternion matrix (time-independent). This paper, by means of complex representation of quaternion matrices, introduces concepts of norms of quaternion matrices, studies the problems of quaternionic Least Squares eigenproblem, and give a practical algebraic technique of computing approximate eigenvalues and eigenvectors of a quaternion matrix in quaternionic quantum mechanics.     

A finite difference continuation method for computing energy levels of Bose-Einstein condensates

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 discretized using the second-order finite difference method (FDM), which is treated as a special case of finite element methods (FEM) using the piecewise bilinear and linear interpolatory functions. Thus the mathematical theory of FEM for elliptic eigenvalue problems (EEP) also holds for the Schrödinger eigenvalue problem (SEP) associated with the GPE. This guarantees the existence of discrete numerical solutions for the ground-state as well as excited-states of the SEP in the variational form. We also study superconvergence of FDM for solution derivatives of parameter-dependent problems (PDP). It is proved that the superconvergence O(ht) in the discrete H¹ norm is achieved, where t=2 and t=1.5 for rectangular and polygonal domains, respectively, and h is the maximal boundary length of difference grids. Moreover, the FDC algorithm can be implemented very efficiently using a simplified two-grid scheme for computing energy levels of the BEC. Numerical results are reported for the ground-state of two-coupled NLS defined in a large square domain, and in particular, for the second-excited state solutions of the 2D BEC in a periodic potential.     

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 compact split step Padé scheme for higher-order nonlinear Schrödinger equation (HNLS) with power law nonlinearity and fourth order dispersion

In this paper we propose a compact split step Padé scheme (CSSPS) to solve the scalar higher-order nonlinear Schrödinger equation (HNLS) with higher-order linear and nonlinear effects such as the third and fourth order dispersion effects, Kerr dispersion, stimulated Raman scattering and power law nonlinearity. The stability of this method has been proved. It has been shown as well that the CSSPS method gives the same results as classical numerical methods like the split step Fourier method and Crank–Nicholson (CN) method but it presents many advantages over theme. It is more efficient. This proposed scheme is well suited to higher-order dispersion effects and readily generalized for nonlinear and dispersion managed fibers. We tested this scheme for the case of the quintic nonlinearity and confirmed that this effect has no significant role on the propagation of single solitons.     

New numerical method for the eigenvalue problem of the 2D Schrödinger equation

The method consists in a flexible transformation of the 2D problem into a set of 1D single and coupled channel problems. This set of problems is then solved numerically by some highly tuned codes. By choosing codes based on CP methods and formulating an ad-hoc shooting procedure for the localization of the eigenenergies we obtain a version which is very efficient for speed and memory requirements. Extension of the method to more dimensions is also possible.     

A new effective algorithm for the resonant state of a Schrödinger equation

In this paper we present a new effective algorithm for the Schrödinger equation. This new method differs from the original Numerov method only in one simple coefficient, by which we can extend the interval of periodicity from 6 to infinity and obtain an embedded correct factor to improve the accuracy. We compare the new method with the original Numerov method by the well-known problem of Woods-Saxon potential. The numerical results show that the new method has great advantage in accuracy over the original. Particularly for the resonant state, the accuracy is improved with four orders overall, and even six to seven orders for the highest oscillatory solution. Surely, this method will replace the original Numerov method and be widely used in various area.     

The global solution for a class of systems of fractional nonlinear Schrödinger equations with periodic boundary condition

In this paper, we consider a class of systems of fractional nonlinear Schrödinger equations. We prove the existence and uniqueness of the global solution to the periodic boundary value problem by using the Faedo-Galërkin method.     

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.     

An efficient split-step compact finite difference method for the coupled Gross–Pitaevskii equations

ABSTRACT

In this study, we propose an efficient split-step compact finite difference (SSCFD) method for computing the coupled Gross–Pitaevskii (CGP) equations. The coupled equations are divided into two parts, nonlinear subproblems and linear ones. Commonly, the nonlinear subproblems could be integrated directly and accurately, but it fails when the time-dependent potential cannot be integrated exactly. In this case, the midpoint and trapezoidal rules are applied approximately. At the same time, the split order is not reduced. For the linear ones, compact finite difference cannot be designed directly. To circumvent this problem, a linear transformation is introduced to decouple the system, which can make the split-step method be used again. Additionally, the proposed SSCFD method also holds for the coupled nonlinear Schrödinger (CNLS) system with time-dependent potential. Finally, numerical experiments for CGP equations and CNLS equations are well simulated, conservative properties and convergence rates are demonstrated as well. It is shown from the numerical tests that the present method is efficient and reliable.