Commun. Comput. Phys., 6 (2009), pp. 639-654.


Explicit Symplectic Methods for the Nonlinear Schrodinger Equation

Hua Guan 1, Yandong Jiao 1, Ju Liu 2, Yifa Tang 1*

1 LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China.
2 Department of Scientific Computation and Applied Software, School of Science, Xi'an Jiaotong University, Xi'an 710049, China.

Received 18 April 2008; Accepted (in revised version) 26 November 2008
Available online 13 February 2009

Abstract

By performing a particular spatial discretization to the nonlinear Schrodinger equation (NLSE), we obtain a non-integrable Hamiltonian system which can be decomposed into three integrable parts (L-L-N splitting). We integrate each part by calculating its phase flow, and develop explicit symplectic integrators of different orders for the original Hamiltonian by composing the phase flows. A 2nd-order reversible constructed symplectic scheme is employed to simulate solitons motion and invariants behavior of the NLSE. The simulation results are compared with a 3rd-order non-symplectic implicit Runge-Kutta method, and the convergence of the formal energy of this symplectic integrator is also verified. The numerical results indicate that the explicit symplectic scheme obtained via L-L-N splitting is an effective numerical tool for solving the NLSE.

AMS subject classifications: 74S30, 65Z05, 65P10, 37M15

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Key words: Explicit symplectic method, L-L-N splitting, nonlinear Schrodinger equation.

*Corresponding author.
Email: guanhua@amss.ac.cn (H. Guan), jiaoyd@lsec.cc.ac.cn (Y. Jiao), liujuy@gmail.com (J. Liu), tyf@lsec.cc.ac.cn (Y. Tang)
 

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