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Hamiltonian Boundary Value Method for the Nonlinear Schrödinger Equation and the Korteweg-de Vries Equation

Part of: Optics, electromagnetic theory - General

Published online by Cambridge University Press:  18 January 2017

Mingzhan Song ,
Xu Qian ,
Hong Zhang  and
Songhe Song
Mingzhan Song*
Affiliation:
College of Science and State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha, Hunan 410073, China
Xu Qian*
Affiliation:
College of Science and State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha, Hunan 410073, China
Hong Zhang*
Affiliation:
College of Science and State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha, Hunan 410073, China
Songhe Song*
Affiliation:
College of Science and State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha, Hunan 410073, China
*
*Corresponding author. Email:smz161619@163.com (M. Song), qianxu@nudt.edu.cn (X. Qian), zhanghnudt@163.com (H. Zhang), shsong@nudt.edu.cn (S. H. Song)
*Corresponding author. Email:smz161619@163.com (M. Song), qianxu@nudt.edu.cn (X. Qian), zhanghnudt@163.com (H. Zhang), shsong@nudt.edu.cn (S. H. Song)
*Corresponding author. Email:smz161619@163.com (M. Song), qianxu@nudt.edu.cn (X. Qian), zhanghnudt@163.com (H. Zhang), shsong@nudt.edu.cn (S. H. Song)
*Corresponding author. Email:smz161619@163.com (M. Song), qianxu@nudt.edu.cn (X. Qian), zhanghnudt@163.com (H. Zhang), shsong@nudt.edu.cn (S. H. Song)
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Abstract

In this paper, we introduce the Hamiltonian boundary value method (HBVM) to solve nonlinear Hamiltonian PDEs. We use the idea of Fourier pseudospectral method in spatial direction, which leads to the finite-dimensional Hamiltonian system. The HBVM, which can preserve the Hamiltonian effectively, is applied in time direction. Then the nonlinear Schrödinger (NLS) equation and the Korteweg-de Vries (KdV) equation are taken as examples to show the validity of the proposed method. Numerical results confirm that the proposed method can simulate the propagation and collision of different solitons well. Meanwhile the corresponding errors in Hamiltonian and other intrinsic invariants are presented to show the good preservation property of the proposed method during long-time numerical calculation.

Keywords

Hamiltonian boundary value methodHamiltonian-preservingnonlinear Schrödinger equationKorteweg-de Vries equation

MSC classification

Secondary: 78A48: Composite media; random media
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 4 , August 2017 , pp. 868 - 886
Copyright
Copyright © Global-Science Press 2017 

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