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Stochastic Multi-Symplectic Integrator for Stochastic Nonlinear Schrödinger Equation

Published online by Cambridge University Press:  03 June 2015

Shanshan Jiang ,
Lijin Wang  and
Jialin Hong
Shanshan Jiang*
Affiliation:
College of Science, Beijing University of Chemical Technology, Beijing 100029, P.R. China
Lijin Wang*
Affiliation:
School of Mathematical Sciences, Graduate University of Chinese Academy of Sciences, Beijing 100049, P.R. China
Jialin Hong*
Affiliation:
State Key Laboratory of Scientific and Engineering Computing, Institute of ComputationalMathematics and Scientific/Engineering Computing, Academy of Mathematics and System Science, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100190, P.R. China
*
Corresponding author.Email:jiangss@mail.buct.edu.cn
Email:ljwang@gucas.ac.cn
Email:hjl@lsec.cc.ac.cn
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Abstract

In this paper we propose stochastic multi-symplectic conservation law for stochastic Hamiltonian partial differential equations, and develop a stochastic multi-symplectic method for numerically solving a kind of stochastic nonlinear Schrödinger equations. It is shown that the stochastic multi-symplectic method preserves the multi-symplectic structure, the discrete charge conservation law, and deduces the recurrence relation of the discrete energy. Numerical experiments are performed to verify the good behaviors of the stochastic multi-symplectic method in cases of both solitary wave and collision.

Keywords

60H1565M0665P10Stochastic nonlinear Schrödinger equationsstochastic multi-symplectic Hamiltonian systemsmulti-symplectic integrators
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 2 , August 2013 , pp. 393 - 411
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Bang, O., Christiansen, P.L., ø, K.Rasmussen, White noise in the two-dimensional nonlinear Schrödinger equation, Appl. Math., 57 (1995), 3–15.Google Scholar
[2]Bouard, A.De, Debussche, A., Di Menza, L., Theoretical and numerical aspects of stochastic nonlinear Schrödinger equations, Monte-Carlo Meth. Appl., 7(1-2)(2001), 55–63.Google Scholar
[3]Debussche, A., Di Menza, L., Numerical simulation of focusing stochastic nonlinear Schrödinger equations, Phys. D,162(2002), 131–154.CrossRefGoogle Scholar
[4]De Bouard, A., Debussche, A., Weak and strong order of convergence of a semi discrete scheme for the stochastic Nonlinear Schrodinger equation, Appl. Mathe. Optim., 54 (2006), 369–399.CrossRefGoogle Scholar
[5]Abdullaev, F.Kh., Garuier, J., Soliton in media with random dispersive perturbations, Physica. D, 134 (1999), 303–315.Google Scholar
[6]Bridges, T., Reich, S., Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A, 284 (2001), 184–193.Google Scholar
[7]Hairer, E., Lubich, C., Wanner, G., Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, Springer-Verlag, 2002.Google Scholar
[8]Hong, J., Liu, Y., H Munthe-Kaas and A Zanna, Globally conservative properties and error estimation of a multi-symplectic scheme for Schrödinger equations with variable coefficients, Appl. Numer. Math., 56 (2006), 814–843.Google Scholar
[9]Hong, J., Liu, X., Li, C., Multi-symplectic Runge-Kutta methods for nonlinear Schrödinger equations with variable coefficients, J. Comput. Phys., 226 (2007), 1968–1984.Google Scholar
[10]Hong, J., Scherer, R., Wang, L., Midpoint Rule for a Linear Stochastic Oscillator with Additive Noise, Neural Parallel and Scientific Computing, 14 (2006), 1–12.Google Scholar
[11]Iserles, A., A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, Cambridge, 1996.Google Scholar
[12]Islas, A., Karpeev, D., Schober, C., Geometric integrators for the nonlinear Schrödinger equation, J. Comput. Phys., 173 (2001), 116–148.Google Scholar
[13]Konotop, V., Vazquez, L., Nonlinear Random Waves, World Scientific, River Edge. NJ, 1994.Google Scholar
[14]Marsden, J., Patrick, G., Shkoller, S., Multi-symplectic geometry, variational integrators, and nonlinear PDEs, Comm. Math. Phys. 199(1998), 351–395.Google Scholar
[15]Milstein, G., Tretyakov, M., Stochastic Numerics for Mathematical Physics, Kluwer Academic Publishers, 1995.Google Scholar
[16]Rasmussen, k. ø., Gaididei, Y.B., Bang, O., Christiansen, P.L., The influence of noise on critical collapse in the nonlinear Schrödinger equation, Phys. Rev. A, 204 (1995), 121–127.Google Scholar
[17]Schober, C., Symplectic integrators for the Ablowitz-Ladik discrete nonlinear Schrödinger equation, Phys. Lett. A, 259 (1999), 140–151.Google Scholar
[18]Shardlow, T., Weak convergence of a numerical method for a stochastic heat equation, BIT, 43 (2003), 179–193.Google Scholar