Li Hongwei, Wu Xiaonan, Zhang Jiwei
School of Mathematical Sciences, Shandong Normal University, Jinan, 250014 People's Republic of China.
Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong, People's Republic of China.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Sep;90(3):033309. doi: 10.1103/PhysRevE.90.033309. Epub 2014 Sep 23.
In this paper, we generalize the unified approach proposed in Zhang et al. [J. Zhang, Z. Xu, and X. Wu, Phys. Rev. E 78, 026709 (2008)] to design the nonlinear local absorbing boundary conditions (LABCs) for the nonlinear Schrödinger equation with wave operator on unbounded domains. In fact, based on the methodology underlying the unified approach, we first split the original equation into two parts-the linear equation and the nonlinear equation-then achieve a one-way operator to approximate the linear equation to make the wave outgoing, and finally combine the one-way operator with the nonlinear equation to achieve the nonlinear LABCs. The stability of the equation with the nonlinear LABCs is also analyzed by introducing some auxiliary variables, and some numerical examples are presented to verify the accuracy and effectiveness of our proposed method.
在本文中，我们推广了Zhang等人[J. Zhang, Z. Xu, and X. Wu, Phys. Rev. E 78, 026709 (2008)]提出的统一方法，以设计无界域上带波动算子的非线性薛定谔方程的非线性局部吸收边界条件（LABCs）。事实上，基于统一方法的基本原理，我们首先将原方程拆分为两部分——线性方程和非线性方程，然后得到一个单向算子来逼近线性方程以使波向外传播，最后将单向算子与非线性方程相结合以得到非线性LABCs。通过引入一些辅助变量，还分析了带有非线性LABCs的方程的稳定性，并给出了一些数值例子来验证我们所提方法的准确性和有效性。
