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

Hamiltonian Boundary Value Method for the Nonlinear Schrödinger Equation and the Korteweg-de Vries Equation
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Year:    2017

Author:    Mingzhan Song, Xu Qian, Hong Zhang, Songhe Song

Advances in Applied Mathematics and Mechanics, Vol. 9 (2017), Iss. 4 : pp. 868–886

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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.2015.m1356

Advances in Applied Mathematics and Mechanics, Vol. 9 (2017), Iss. 4 : pp. 868–886

Published online:    2017-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    19

Keywords:    Hamiltonian boundary value method Hamiltonian-preserving nonlinear Schrödinger equation Korteweg-de Vries equation.

Author Details

Mingzhan Song

Xu Qian

Hong Zhang

Songhe Song

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