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带乘性噪声的空间分数阶随机非线性Schrödinger方程的广义多辛算法

刘子源, 梁家瑞, 钱旭, 宋松和   

  1. 国防科技大学数学系, 长沙 410073
  • 收稿日期:2019-05-10 出版日期:2019-12-15 发布日期:2019-11-16
  • 基金资助:

    国家自然科学基金(11571366),湖南省自然科学基金(S2017JJQNJJ0764)和国防科技大学科研计划项目(ZK17-03-27)资助项目.

刘子源, 梁家瑞, 钱旭, 宋松和. 带乘性噪声的空间分数阶随机非线性Schrödinger方程的广义多辛算法[J]. 计算数学, 2019, 41(4): 440-452.

Liu Ziyuan, Liang Jiarui, Qian Xu, Song Songhe. A GENERALIZED MULTI-SYMPLECTIC METHOD FOR STOCHASTIC SPACE-FRACTIONAL NONLINEAR SCHRÖDINGER EQUATION WITH MULTIPLICATIVE NOISE[J]. Mathematica Numerica Sinica, 2019, 41(4): 440-452.

A GENERALIZED MULTI-SYMPLECTIC METHOD FOR STOCHASTIC SPACE-FRACTIONAL NONLINEAR SCHRÖDINGER EQUATION WITH MULTIPLICATIVE NOISE

Liu Ziyuan, Liang Jiarui, Qian Xu, Song Songhe   

  1. Department of mathematics, National University of Defense Technology, Changsha 410073, China
  • Received:2019-05-10 Online:2019-12-15 Published:2019-11-16
带乘性噪声的空间分数阶随机非线性Schrödinger方程是一类重要的方程,可应用于描述开放非局部量子系统的演化过程.该方程为一个无穷维分数阶随机Hamilton系统,且具有广义多辛结构和质量守恒的性质.针对该方程的广义多辛形式,在空间上采用拟谱方法离散分数阶微分算子,在时间上则采用隐式中点格式,构造出一类保持全局质量的广义多辛格式.对行波解和平面波解等进行数值模拟,结果验证了所构造格式的有效性和保结构性质,时间均方收敛阶约在0.5到1之间.
The stochastic space-fractional nonlinear Schrödinger equation with multiplicative noise is an important equation which describes the evolution of an open nonlocal quantum system. In this paper we prove that this system is an infinite-dimensional stochastic fractional Hamiltonian system, and satisfies both the mass and the generalized multi-symplectic conservation law. After that, with the Fourier pseudo-spectral approximation to the spatial fractional Laplacian operator and the implicit mid-point method for time discretization, we propose a mass-conserving generalized stochastic multi-symplectic method. Numerical simulations are presented for soliton solution and plane wave solution. The results demonstrate the effectiveness and conservative property of the proposed methods. Furthermore, the results show that the mean square convergence order on time is approximately 0.5 to 1.

MR(2010)主题分类: 

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