Volume 9, Issue 4
Hamiltonian Boundary Value Method for the Nonlinear Schrödinger Equation and the Korteweg-de Vries Equation

Adv. Appl. Math. Mech., 9 (2017), pp. 868-886.

Published online: 2018-05

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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 method, Hamiltonian-preserving, nonlinear Schrödinger equation, Korteweg-de Vries equation.

65M10, 78A48

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@Article{AAMM-9-868, author = {Mingzhan Song , and Xu Qian , and Hong Zhang , and Song , Songhe}, title = {Hamiltonian Boundary Value Method for the Nonlinear Schrödinger Equation and the Korteweg-de Vries Equation}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {9}, number = {4}, pages = {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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2015.m1356}, url = {http://global-sci.org/intro/article_detail/aamm/12180.html} }
TY - JOUR T1 - Hamiltonian Boundary Value Method for the Nonlinear Schrödinger Equation and the Korteweg-de Vries Equation AU - Mingzhan Song , AU - Xu Qian , AU - Hong Zhang , AU - Song , Songhe JO - Advances in Applied Mathematics and Mechanics VL - 4 SP - 868 EP - 886 PY - 2018 DA - 2018/05 SN - 9 DO - http://doi.org/10.4208/aamm.2015.m1356 UR - https://global-sci.org/intro/article_detail/aamm/12180.html KW - Hamiltonian boundary value method, Hamiltonian-preserving, nonlinear Schrödinger equation, Korteweg-de Vries equation. AB -

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.

Mingzhan Song, Xu Qian, Hong Zhang & Songhe Song. (2020). Hamiltonian Boundary Value Method for the Nonlinear Schrödinger Equation and the Korteweg-de Vries Equation. Advances in Applied Mathematics and Mechanics. 9 (4). 868-886. doi:10.4208/aamm.2015.m1356
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