Stability Analysis for Nonlinear Schrödinger Equations with Nonlinear Absorbing Boundary Conditions
Jiwei Zhang 1, Zhenli Xu 2, Xiaonan Wu 3, Desheng Wang 41 Applied and Computational Mathematics Division, Beijing Computational Science Research Center, China
2 Department of Mathematics, Institute of Natural Sciences, and MOE Key Lab of Scientific and Engineering Computing, Shanghai Jiao Tong University, Shanghai 200240, China
3 Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong, China
4 Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
Received 2014-4-7 Accepted 2016-8-28
Available online 2017-1-18
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.
Key words: Nonlinear Schrödinger equations, Energy estimates, Absorbing boundary conditions.
AMS subject classifications: 65M12, 65M06, 65M15.
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