基于广义有限差分方法计算二维非线性耦合薛定谔方程
Calculating the Two-Dimensional Nonlinear Coupled Schrödinger Equation Based on the Generalized Finite Difference Method
摘要: 耦合非线性薛定谔方程组(Coupled Nonlinear Schrödinger Equations, CNLS)描述了许多量子系统的动力学行为，寻找一种有效的数值方法是科学和工程应用的关键。本文介绍了一种求解二维非线性耦合薛定谔方程的无网格方法，该方法称为广义有限差分法(GFDM)。该方法将Taylor多项式与最小二乘法相结合，在推导过程中，首先将CNLS非线性系统在时间方向使用线性化方法进行离散，推导出CNLS在时间上的线性化离散形式，再使用GFDM在空间上的计算。该方法为CNLS生成稀疏的非线性代数系统。最后给出了数值算例，表明了本文方法求解两个区域线性代数方程的有效性。数值结果表明，发展的数值方法是一种稳定、快速和精确的计算方法。
Abstract: Coupled nonlinear Schrödinger equations (CNLS) describe the dynamical behavior of many quantum systems, and finding an efficient numerical method is crucial for scientific and engineering applications. This paper introduces a meshless method for solving two-dimensional nonlinear coupled Schrödinger equations, called the generalized finite difference method (GFDM). This method combines Taylor polynomials with the least squares method. In the derivation, the CNLS nonlinear system is first discretized in the time direction using the Crank-Nicolson method, deriving the linearized discretized form of CNLS in time. Then, GFDM is used for spatial computation. This method generates a sparse nonlinear algebraic system for CNLS. Finally, numerical examples are given to demonstrate the effectiveness of the proposed method in solving linear algebraic equations in two regions. Numerical results show that the developed numerical method is a stable, fast, and accurate computational approach.
文章引用：陈贝贝, 纪翠翠. 基于广义有限差分方法计算二维非线性耦合薛定谔方程[J]. 应用数学进展, 2026, 15(2): 80-93. https://doi.org/10.12677/aam.2026.152051

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