Stability Analysis for Nonlinear Schrödinger Equations with Nonlinear Absorbing Boundary Conditions

Authors

  • Jiwei Zhang Beijing Computational Science Research Center, Beijing 10084, P.R.China
  • Zhenli Xu School of Mathematical Sciences, Institute of Natural Sciences and Key Laboratory of Scientific and Engineering Computing (Ministry of Education), Shanghai Jiao Tong University, Shanghai 200240, China
  • Xiaonan Wu Department of Mathematics, Hong Kong Baptist University, Hong Kong, P.R. China
  • Desheng Wang Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore

DOI:

https://doi.org/10.4208/jcm.1608-m4507

Keywords:

Nonlinear Schrödinger equations, Energy estimates, Absorbing boundary conditions.

Abstract

Local absorbing boundary conditions (LABCs) for nonlinear Schrödinger equations have been constructed in papers [PRE 78(2008) 026709; and PRE 79 (2009) 046711] using the so-called unified approach. In this paper, we present stability analysis for the reduced problem with LABCs on the bounded computational domain by the energy estimate, and discuss a class of modified versions of LABCs. To prove the stability analysis of the reduced problem, we turn to the technique of some auxiliary variables which reduces the higher-order derivatives in LABCs into a family of equations with lower-order derivatives. Furthermore, we extend the strategy to the stability analysis of two-dimensional problems by carefully dealing with the LABCs at corners. Numerical examples are given to demonstrate the effectiveness of our boundary conditions and validate the theoretical analysis.

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Published

2018-08-22

Issue

Vol. 35 No. 1 (2017)

Section

Articles