J. Comp. Math., 35 (2017), pp. 1-18.


Stability Analysis for Nonlinear Schrödinger Equations with Nonlinear Absorbing Boundary Conditions

Jiwei Zhang 1, Zhenli Xu 2, Xiaonan Wu 3, Desheng Wang 4

1 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
doi:10.4208/jcm.1608-m4507

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.

Key words: Nonlinear Schrödinger equations, Energy estimates, Absorbing boundary conditions.

AMS subject classifications: 65M12, 65M06, 65M15.


Email: jwzhang@csrc.ac.cn, xuzl@sjtu.edu.cn, xwu@hkbu.edu.hk, desheng@ntu.edu.sg
 

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