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Volume 6, Issue 3
Explicit Symplectic Methods for the Nonlinear Schrödinger Equation

Hua Guan, Yandong Jiao, Ju Liu & Yifa Tang

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

Published online: 2009-06

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  • Abstract

By performing a particular spatial discretization to the nonlinear Schrödinger 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.

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@Article{CiCP-6-639, author = {Hua Guan, Yandong Jiao, Ju Liu and Yifa Tang}, title = {Explicit Symplectic Methods for the Nonlinear Schrödinger Equation}, journal = {Communications in Computational Physics}, year = {2009}, volume = {6}, number = {3}, pages = {639--654}, abstract = {

By performing a particular spatial discretization to the nonlinear Schrödinger 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.

}, issn = {1991-7120}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cicp/7698.html} }
TY - JOUR T1 - Explicit Symplectic Methods for the Nonlinear Schrödinger Equation AU - Hua Guan, Yandong Jiao, Ju Liu & Yifa Tang JO - Communications in Computational Physics VL - 3 SP - 639 EP - 654 PY - 2009 DA - 2009/06 SN - 6 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cicp/7698.html KW - AB -

By performing a particular spatial discretization to the nonlinear Schrödinger 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.

Hua Guan, Yandong Jiao, Ju Liu and Yifa Tang. (2009). Explicit Symplectic Methods for the Nonlinear Schrödinger Equation. Communications in Computational Physics. 6 (3). 639-654. doi:
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