School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, People's Republic of China.
Phys Rev E. 2017 Dec;96(6-1):063305. doi: 10.1103/PhysRevE.96.063305. Epub 2017 Dec 13.
The numerical solution of the general coupled nonlinear Schrödinger equations on unbounded domains is considered by applying the artificial boundary method in this paper. In order to design the local absorbing boundary conditions for the coupled nonlinear Schrödinger equations, we generalize the unified approach previously proposed [J. Zhang et al., Phys. Rev. E 78, 026709 (2008)PLEEE81539-375510.1103/PhysRevE.78.026709]. Based on the methodology underlying the unified approach, the original problem is split into two parts, linear and nonlinear terms, and we then achieve a one-way operator to approximate the linear term to make the wave out-going, and finally we combine the one-way operator with the nonlinear term to derive the local absorbing boundary conditions. Then we reduce the original problem into an initial boundary value problem on the bounded domain, which can be solved by the finite difference method. The stability of the reduced problem is also analyzed by introducing some auxiliary variables. Ample numerical examples are presented to verify the accuracy and effectiveness of our proposed method.
本文应用人工边界法研究了无界区域上一般耦合非线性薛定谔方程组的数值解。为了设计耦合非线性薛定谔方程组的局部吸收边界条件，我们推广了先前提出的统一方法[J. Zhang 等人，物理评论 E78, 026709（2008）PLEEE81539-375510.1103/PhysRevE.78.026709]。基于统一方法的基本原理，将原始问题分解为线性项和非线性项两部分，然后我们通过一个单向算子来近似线性项，使得波向外传播，最后将单向算子与非线性项相结合，推导出局部吸收边界条件。然后，我们将原始问题简化为有界区域上的初边值问题，可以用有限差分法求解。通过引入一些辅助变量，分析了简化问题的稳定性。本文还给出了大量数值算例，验证了所提出方法的准确性和有效性。
