Commun. Comput. Phys., 12 (2012), pp. 1392-1416.


Numerical Solutions of Coupled Nonlinear Schrodinger Equations by Orthogonal Spline Collocation Method

Qing-Jiang Meng 1*, Li-Ping Yin 2, Xiao-Qing Jin 1, Fang-Li Qiao 3

1 Department of Mathematics, University of Macau, Macao.
2 First Institute of Oceanography, State Oceanic Administration, Qingdao, Shandong 266061, China; College of Physical and Environmental Ocanography, Ocean University of China, Qingdao, Shandong 266003, China.
3 Key Laboratory of Marine Science and Numerical Modeling of State Oceanic Administration and First Institute of Oceanography, State Oceanic Administration, Qingdao, Shandong 266061, China.

Received 18 April 2011; Accepted (in revised version) 9 January 2012
Available online 8 May 2012
doi:10.4208/cicp.180411.090112a

Abstract

In this paper, we present the use of the orthogonal spline collocation method for the semi-discretization scheme of the one-dimensional coupled nonlinear Schrodinger equations. This method uses the Hermite basis functions, by which physical quantities are approximated with their values and derivatives associated with Gaussian points. The convergence rate with order $O(h^4+\tau^2)$ and the stability of the scheme are proved. Conservation properties are shown in both theory and practice. Extensive numerical experiments are presented to validate the numerical study under consideration.

AMS subject classifications: 65N35, 35C45, 35L65

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Key words: Coupled nonlinear Schrodinger equations, orthogonal spline collocation method, conservation law.

*Corresponding author.
Email: qjmeng04@163.com (Q.-J. Meng), lipyconstance@163.com (L.-P. Yin), xqjin@umac.mo (X.-Q. Jin), qiaofl@fio.org.cn (F.-L. Qiao)
 

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