Numerical simulation of coupled nonlinear Schrödinger equation

The coupled nonlinear Schrödinger equation models several interesting physical phenomena. It represents a model equation for optical fiber with linear birefringence. In this paper we introduce a finite difference method for a numerical simulation of this equation. This method is second-order in space and conserves the energy exactly. It is quite accurate and describes the interaction picture clearly according to our numerical results.     

Numerical solution of two-dimensional Schrödinger equation by Boadway's transformation

In this article, Boadway's transformation technique is extended to the two-dimensional Schrödinger equation. The result of numerical experiments is presented, and a comparison of the present results with the exact solution shows an extremely good agreement.     

A method for the solution of the Schrödinger equation

A method has been developed for the numerical calculation of eigenvalues of the Schrödinger equation. The eigenvalues are computed directly as roots of a function known in transmission line theory as the impedance. The novel numerical algorithm is simple and very effective in calculating the eigenvalues and eigenfunctions.     

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.     

New schemes for the coupled nonlinear Schrödinger equation

In this paper, we present three new schemes for the coupled nonlinear Schrödinger equation. The three new schemes are multi-symplectic schemes that preserve the intrinsic geometry property of the equation. The three new schemes are also semi-explicit in the sense that they need not solve linear algebraic equations every time-step, which is usually the most expensive in numerical simulation of partial differential equations. Many numerical experiments on collisions of solitons are presented to show the efficiency of the new multi-symplectic schemes.     

Existence of the global solution for fractional logarithmic Schrödinger equation

In this paper we consider the initial boundary value problem for a class of fractional logarithmic Schrödinger equation. By using the fractional logarithmic Sobolev inequality and introducing a family of potential wells, we give some properties of the family of potential wells and obtain existence of global solution.     

Dissipative exponentially-fitted methods for the numerical solution of the Schrödinger equation

The first dissipative exponentially fitted method for the numerical integration of the Schr?dinger equation is developed in this paper. The technique presented is a nonsymmetric multistep (dissipative) method. An application to the bound-states problem and the resonance problem of the radial Schr?dinger equation indicates that the new method is more efficient than the classical dissipative method and other well-known methods. Based on the new method and the method of Raptis and Allison (Comput. Phys. Commun. 14 (1978) 1-5) a new variable-step method is obtained. The application of the new variable-step method to the coupled differential equations arising from the Schr?dinger equation indicates the power of the new approach.     

On finite difference methods for the solution of the Schrödinger equation

A review for the numerical methods used for the solution of the Schrödinger equation is presented.     

An expert system for the numerical solution of the radial Schrödinger equation

An expert system for the numerical solution of the phase shift problem of the radial Schrödinger equation is developed in this paper.     

Numerical analysis of a new conservative scheme for the coupled nonlinear Schrödinger equations

We devote the present paper to an efficient conservative scheme for the coupled nonlinear Schrödinger (CNLS) system, based on the Fourier pseudospectral method, the Crank–Nicolson method and leap-frog method. To obtain the present scheme, the key idea consists of two aspects. First, we solve the CNLS system based on its Hamiltonian structure and the resulted scheme can preserve the Hamiltonian nature. Second, we use Fourier pseudospectral method in spatial discretization and Crank–Nicolson/ leap-frog scheme for discretizing linear/ nonlinear terms in time direction, respectively. The proposed scheme is energy-preserving, mass-preserving, uniquely solvable and unconditionally stable, while being decoupled, linearized and suitable for parallel computation in practical computation. Using the energy method and the classical interpolation theory, an error estimate of the proposed scheme is proven strictly without any grid ratio restrictions in the discrete L² norm. Finally, numerical results are reported to verify our theoretical analysis.     

Periodic and solitary wave solutions of the attractive and repulsive nonlinear Schrödinger equation

By means of symbolic computation, the first integral method is presented to obtain novel exact solutions of the nonlinear evolution equation. The obtained results include periodic and solitary wave solutions. The power of this manageable method is confirmed and the availability of symbolic computation is demonstrated.     

Dissipative high phase-lag order Numerov-type methods for the numerical solution of the Schrödinger equation

A family of explicit Numerov-type methods with minimal phase-lag are constructed in this paper. These methods are of algebraic order six and have phase-lag order 10(2)26. The main characteristic of these methods is that they are dissipative, i.e. they are not symmetric and they have no interval of periodicity. Numerical results indicate that these new methods are more efficient than older ones, i.e. the property of the phase-lag order is more crucial than a non-empty interval of periodicity for the construction of numerical methods for the solution of the Schrödinger type equations.     

On variable-step methods for the numerical solution of Schrödinger equation and related problems

In this paper we present a review for the construction of variable-step methods for the numerical integration of the Schr?dinger equation. Phase-lag and stability are investigated. The methods are variable-step because of a simple natural error control mechanism. Numerical results obtained for coupled differential equations arising from the Schr?dinger equation and for the wave equation show the validity of the approach presented.     

An OpenCL implementation for the solution of the time-dependent Schrödinger equation on GPUs and CPUs

Open Computing Language (OpenCL) is a parallel processing language that is ideally suited for running parallel algorithms on Graphical Processing Units (GPUs). In the present work we report on the development of a generic parallel single-GPU code for the numerical solution of a system of first-order ordinary differential equations (ODEs) based on the OpenCL model. We have applied the code in the case of the Time-Dependent Schrödinger Equation of atomic hydrogen in a strong laser field and studied its performance on NVIDIA and AMD GPUs against the serial performance on a CPU. We found excellent scalability and a significant speedup of the GPU over the CPU device. The speedup in the benchmark tended towards a value of about 40 with significant speedups expected against multi-core CPUs. Furthermore, though we do not present the detailed benchmarks here, we also have achieved speedup values of around 75 by performing a slight optimization of the described algorithm.     

A conservative exponential time differencing method for the nonlinear cubic Schrödinger equation

A mass and energy conservative exponential time differencing scheme using the method of lines is proposed for the numerical solution of a certain family of first-order time-dependent PDEs. The resulting nonlinear system is solved with an unconditionally stable modified predictor–corrector method using a second-order explicit scheme. The efficiency of the method introduced is analyzed and discussed by applying it to the nonlinear cubic Schrödinger equation. The results arising from the experiments for the single, the double soliton waves and the system of two Schrödinger equations are compared with relevant known ones.