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.     

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.     

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.     

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 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 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.     

An efficient and accurate quantum lattice-gas model for the many-body Schrödinger wave equation

Presented is quantum lattice-gas model for simulating the time-dependent evolution of a many-body quantum mechanical system of particles governed by the non-relativistic Schrödinger wave equation with an external scalar potential. A variety of computational demonstrations are given where the numerical predictions are compared with exact analytical solutions. In all cases, the model results accurately agree with the analytical predictions and we show that the model's error is second order in the temporal discretization and fourth order in the spatial discretization. The difficult problem of simulating a system of fermionic particles is also treated and a general computational formulation of this problem is given. For pedagogical purposes, the two-particle case is presented and the numerical dispersion of the simulated wave packets is compared with the analytical solutions.     

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.     

Domain wall arrays, fronts, and bright and dark solitons in a generalized derivative nonlinear Schrödinger equation

We obtain some exact solutions of a generalized derivative nonlinear Schrödinger equation, including domain wall arrays (periodic solutions in terms of elliptic functions), fronts, and bright and dark solitons. In certain parameter domains, fundamental bright and dark solitons are chiral, and the propagation direction is determined by the sign of the self-steepening parameter. Moreover, we also find the chirping reversal phenomena of fronts, and bright and dark solitons, and discuss two different ways to produce the chirping reversal.     

The accurate numerical solution of the Schrödinger equation with an explicitly time-dependent Hamiltonian

We show how the highly accurate and efficient Constant Perturbation (CP) technique for steady-state Schrödinger problems can be used in the solution of time-dependent Schrödinger problems with explicitly time-dependent Hamiltonians, following a technique suggested by Ixaru (2010). By introducing a sectorwise spatial discretization using bases of accurately CP-computed eigenfunctions of carefully-chosen stationary problems, we deal with the possible highly oscillatory behavior of the wave function while keeping the dimension of the resulting ODE system low. Also for the time-integration of the ODE system a very effective CP-based approach can be used.     

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.     

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.     

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.     

A symmetric eight-step predictor-corrector method for the numerical solution of the radial Schrödinger equation and related IVPs with oscillating solutions

In this paper we present a new optimized symmetric eight-step predictor-corrector method with phase-lag of order infinity (phase-fitted). The method is based on the symmetric multistep method of Quinlan–Tremaine, with eight steps and eighth algebraic order and is constructed to solve numerically the radial time-independent Schrödinger equation during the resonance problem with the use of the Woods–Saxon potential. It can also be used to integrate related IVPs with oscillating solutions such as orbital problems. We compare the new method to some recently constructed optimized methods from the literature. We measure the efficiency of the methods and conclude that the new method with infinite order of phase-lag is the most efficient of all the compared methods and for all the problems solved.     

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.     

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.     

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.     

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 methods for the coupled 1D nonlinear Schrödinger system

In the paper, the multi-symplectic formulation of the coupled 1D nonlinear Schrödinger system (CNLS) is considered. For the multi-symplectic formulation, a new six point scheme, which is equivalent to the multi-symplectic Preissman integrator, is derived. We also present numerical experiments, which show that the multi-symplectic scheme has excellent long-time numerical behaviour and energy conservation property.