Abstract

On the basis of sampling approximation for a function defined on a bounded interval by combining Coiflet-type wavelet expansion and technique of boundary extension, a space-time fully decoupled formulation is proposed to solve multidimensional Schrödinger equations with generalized nonlinearities and damping. By applying a wavelet Galerkin approach for spatial discretization, nonlinear Schrödinger equations are first transformed into a system of ordinary differential equations, in which all matrices are completely independent of time and never need to be recalculated in the time integration. Then, the classical fourth-order explicit Runge–Kutta method is used to solve the resulting semidiscretization system. By studying several widely considered test problems, results demonstrate that when a relatively fine mesh is adopted, the present wavelet algorithm has a much better computational accuracy and efficiency than many existing numerical methods, due to its higher order of convergence in space which can go up to 6.

1. Introduction

The time-dependent Schrödinger equation with nonlinear potential functions is one of the most important mathematical models in modern science due to its wide applications in many different fields, such as nonlinear optics, plasma physics, and semiconductor industry [1–5]. The present study considers the three-dimensional generalized nonlinear Schrödinger equation (GNLSE) [6–37]with the initial conditionand the Dirichlet boundary conditionIn (1)–(3), is the complex unit, with vector is a complex-valued unknown function, is the Laplace operator, and , , , and are known real functions for and . In GNLSE (1), the external potential is a given real function, whose specific form is dependent on different problems. For example, it is usually chosen as either a harmonic confining potential , or an optical lattice potential with constants , , and , for studying the Bose–Einstein condensation [5, 26]. The nonlinearities and which are real-valued smooth functions with respect to the density are determined by the specific application. Note that, in this study, the most popular nonlinear term with constant [1–10] has been extended to the general form which can be chosen as arbitrary function as needed, such as , , , and , where , and are real constants [26, 36]. Moreover, the general damping term which is usually ignored in many studies [8–16, 18–25, 27–34] is also considered in GNLSE (1), because the damping effect may play an important role and should not be ignored in some physical processes [7, 17, 26, 35, 36], such as the inelastic collisions in Bose–Einstein condensation [7, 26].

Because of the broad applications of GNLSE (1), developing accurate and efficient numerical methods for solving such equations has attracted considerable research attention in the past few years. For instance, Mohebbi and Dehghan [15] studied two-dimensional linear Schrödinger equations by using a compact boundary value method with fourth-order accuracy in both space and time. Liao et al. [16] applied a fourth-order compact difference scheme to solve the two-dimensional linear Schrödinger equation without damping term. In order to save cost on computation, the finite difference schemes combining with the alternating direction implicit method are developed to study the multidimensional Schrödinger equations [18–20, 23, 24, 30, 34]. In such algorithms, handling multidimensional problems is transformed into solving a series of one-dimensional problems by introducing intermediate variables. Moreover, many other numerical methods are also proposed to solve multidimensional Schrödinger equations, such as the collocation method [21, 22, 27, 33], the Galerkin method [27, 31], and the mesh-free methods [25, 31, 32]. The above methods are effective for solving Schrödinger equations under certain conditions. However, many of them will encounter severe difficulties in uniformly solving the three-dimensional generalized nonlinear Schrödinger equation (1). For example, the time-splitting methods need to obtain the density by solving analytically a nonlinear differential equation [7, 8, 21, 26], which is an extremely difficult task for general damping term . And when the classical collocation method [25] and Galerkin type method [32] are employed to solve directly the nonlinear Schrödinger equation, the matrices generated in the spatial discretization of nonlinear terms will be dependent on the time-dependent unknown vector and must be recalculated at each time step [25, 32, 38], thereby consuming considerable computing resources. Because such repeated recalculations of matrices from the spatial discretization of nonlinear term can be regarded as reperforming the spatial discretization at each time step [38], these methods [25, 27, 32] cannot divide the solution procedure into two completely separate processes, that is, the spatial discretization and the time integration. Therefore, the decoupling between spatial and temporal discretizations in these methods [25, 27, 32] is incomplete [38].

In the current work, a space-time fully decoupled formulation by combining a wavelet Galerkin technique and the classical fourth-order explicit Runge–Kutta method is proposed to uniformly solve the three-dimensional generalized nonlinear Schrödinger equation (1). In such a space-time fully decoupled wavelet formulation, all matrices generated in the spatial discretization of the nonlinear partial differential equation (1) are constant matrices and need not to be updated in the subsequent time integration. In addition, a systematic comparison between the present solutions and those obtained by using many existing numerical methods is conducted by solving several widely considered test problems.

2. Sampling Approximation of an Interval-Bounded -Function

Based on the theory of wavelet based multiresolution analysis, a set of scaling bases for three-dimensional space can be directly obtained by the tensor products of one-dimensional wavelet bases [39, 40]. Therefore following our previous work [41–44], a function can be approximated aswhere , , and are the decomposition level, respectively, in the , , and directions, and the modified one-dimensional wavelet basis [41–44]withIn (5), is the generalized Coiflet-type orthogonal scaling function with first-order moment and number of vanishing moments of the corresponding wavelet function, which is developed by Wang [41]. In addition, the functionsin which coefficients and of numerical differentiation are determined by matrices ,  , , and with subscripts .

Since the sampling approximation (4) is valid for all functions , for arbitrary nonlinear operator holding , by treating as a new function and applying (4), we havewhich provides the foundation for conducting a space-time fully decoupled formulation [38].

3. Solution of the Generalized Nonlinear Schrödinger Equation

At the beginning of the solution to the three-dimensional generalized nonlinear Schrödinger equation (1), by introducing , where and are real-valued functions, we rewrite it into two coupled time-dependent nonlinear partial differential equationsin which and .

Following the wavelet approximations (4) and (8), the unknown functions , nonlinear terms and , and terms and can be expressed as

Substituting (10)–(14) into (9), then multiplying both sides of the resulting equation by ,  ,  , and , respectively, and performing integration over the domain , one can obtain a series of ordinary differential equations. Considering boundary conditions (3), these ordinary differential equations can be written intoin which vectors , , , matrices , , and subscripts , , , , , , and . In (15) and (16), vector operational rules and will always hold for vectors and , and the generalized connection coefficient can be exactly and readily obtained without numerical integral but based on the database independent of the specific problems [38]. In addition, the vector determined by boundary conditions can be expressed aswith