Multi-symplectic variational integrators for nonlinear Schrödinger equations with variable coefficients

doi:10.1088/1674-1056/25/1/010205

中国物理B ›› 2016, Vol. 25 ›› Issue (1): 10205-010205.doi: 10.1088/1674-1056/25/1/010205

Multi-symplectic variational integrators for nonlinear Schrödinger equations with variable coefficients

Cui-Cui Liao(廖翠萃), Jin-Chao Cui(崔金超), Jiu-Zhen Liang(梁久祯), Xiao-Hua Ding(丁效华)

1. Department of Information and Computing Science, College of Science, Jiangnan University, Wuxi 214122, China;
2. College of Internet of Things, Jiangnan University, Wuxi 214122, China;
3. Department of Mathematics, Harbin Institute of Technology. Harbin 150001, China

收稿日期:2015-07-08 修回日期:2015-09-18 出版日期:2016-01-05 发布日期:2016-01-05
通讯作者: Cui-Cui Liao E-mail:cliao@jiangnan.edu.cn
基金资助:
Project supported by the National Natural Science Foundation of China (Grant No. 11401259) and the Fundamental Research Funds for the Central Universities, China (Grant No. JUSRR11407).

Multi-symplectic variational integrators for nonlinear Schrödinger equations with variable coefficients

Cui-Cui Liao(廖翠萃)¹, Jin-Chao Cui(崔金超)¹, Jiu-Zhen Liang(梁久祯)², Xiao-Hua Ding(丁效华)³

1. Department of Information and Computing Science, College of Science, Jiangnan University, Wuxi 214122, China;
2. College of Internet of Things, Jiangnan University, Wuxi 214122, China;
3. Department of Mathematics, Harbin Institute of Technology. Harbin 150001, China

Received:2015-07-08 Revised:2015-09-18 Online:2016-01-05 Published:2016-01-05
Contact: Cui-Cui Liao E-mail:cliao@jiangnan.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (Grant No. 11401259) and the Fundamental Research Funds for the Central Universities, China (Grant No. JUSRR11407).

摘要/Abstract

摘要： In this paper, we propose a variational integrator for nonlinear Schrödinger equations with variable coefficients. It is shown that our variational integrator is naturally multi-symplectic. The discrete multi-symplectic structure of the integrator is presented by a multi-symplectic form formula that can be derived from the discrete Lagrangian boundary function. As two examples of nonlinear Schrödinger equations with variable coefficients, cubic nonlinear Schrödinger equations and Gross-Pitaevskii equations are extensively studied by the proposed integrator. Our numerical simulations demonstrate that the integrator is capable of preserving the mass, momentum, and energy conservation during time evolutions. Convergence tests are presented to verify that our integrator has second-order accuracy both in time and space.

关键词: multi-symplectic form formulas, variational integrators, conservation laws, nonlinear Schrö, dinger equations

Abstract: In this paper, we propose a variational integrator for nonlinear Schrödinger equations with variable coefficients. It is shown that our variational integrator is naturally multi-symplectic. The discrete multi-symplectic structure of the integrator is presented by a multi-symplectic form formula that can be derived from the discrete Lagrangian boundary function. As two examples of nonlinear Schrödinger equations with variable coefficients, cubic nonlinear Schrödinger equations and Gross-Pitaevskii equations are extensively studied by the proposed integrator. Our numerical simulations demonstrate that the integrator is capable of preserving the mass, momentum, and energy conservation during time evolutions. Convergence tests are presented to verify that our integrator has second-order accuracy both in time and space.

Key words: multi-symplectic form formulas, variational integrators, conservation laws, nonlinear Schrödinger equations

中图分类号: (Numerical simulation; solution of equations)

02.60.Cb

02.70.Bf (Finite-difference methods) 45.10.Na (Geometrical and tensorial methods) 47.10.Df (Hamiltonian formulations)

廖翠萃, 崔金超, 梁久祯, 丁效华. Multi-symplectic variational integrators for nonlinear Schrödinger equations with variable coefficients[J]. 中国物理B, 2016, 25(1): 10205-010205.

Cui-Cui Liao(廖翠萃), Jin-Chao Cui(崔金超), Jiu-Zhen Liang(梁久祯), Xiao-Hua Ding(丁效华). Multi-symplectic variational integrators for nonlinear Schrödinger equations with variable coefficients[J]. Chin. Phys. B, 2016, 25(1): 10205-010205.

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Multi-symplectic variational integrators for nonlinear Schrödinger equations with variable coefficients

Multi-symplectic variational integrators for nonlinear Schrödinger equations with variable coefficients

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