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A Compact Scheme for Coupled Stochastic Nonlinear Schrödinger Equations

Part of: Stochastic analysis Infinite-dimensional Hamiltonian systems

Published online by Cambridge University Press:  05 December 2016

Chuchu Chen ,
Jialin Hong ,
Lihai Ji  and
Linghua Kong
Chuchu Chen*
Affiliation:
Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Jialin Hong*
Affiliation:
Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Lihai Ji*
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing 100094, China
Linghua Kong*
Affiliation:
School of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, China
*
*Corresponding author. Email addresses:chenchuchu@lsec.cc.ac.cn (C. Chen), hjl@lsec.cc.ac.cn (J. Hong), jilihai@lsec.cc.ac.cn (L. Ji), konglh@mail.ustc.edu.cn (L. Kong)
*Corresponding author. Email addresses:chenchuchu@lsec.cc.ac.cn (C. Chen), hjl@lsec.cc.ac.cn (J. Hong), jilihai@lsec.cc.ac.cn (L. Ji), konglh@mail.ustc.edu.cn (L. Kong)
*Corresponding author. Email addresses:chenchuchu@lsec.cc.ac.cn (C. Chen), hjl@lsec.cc.ac.cn (J. Hong), jilihai@lsec.cc.ac.cn (L. Ji), konglh@mail.ustc.edu.cn (L. Kong)
*Corresponding author. Email addresses:chenchuchu@lsec.cc.ac.cn (C. Chen), hjl@lsec.cc.ac.cn (J. Hong), jilihai@lsec.cc.ac.cn (L. Ji), konglh@mail.ustc.edu.cn (L. Kong)
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Abstract

In this paper, we propose a compact scheme to numerically study the coupled stochastic nonlinear Schrödinger equations. We prove that the compact scheme preserves the discrete stochastic multi-symplectic conservation law, discrete charge conservation law and discrete energy evolution law almost surely. Numerical experiments confirm well the theoretical analysis results. Furthermore, we present a detailed numerical investigation of the optical phenomena based on the compact scheme. By numerical experiments for various amplitudes of noise, we find that the noise accelerates the oscillation of the soliton and leads to the decay of the solution amplitudes with respect to time. In particular, if the noise is relatively strong, the soliton will be totally destroyed. Meanwhile, we observe that the phase shift is sensibly modified by the noise. Moreover, the numerical results present inelastic interaction which is different from the deterministic case.

Keywords

Coupled stochastic nonlinear Schrödinger equationscompact schemestochastic multi-symplectic conservation lawenergy evolution lawcharge conservation lawsoliton evolutionsoliton interaction

MSC classification

Secondary: 60H15: Stochastic partial differential equations 60H35: Computational methods for stochastic equations 37K05: Hamiltonian structures, symmetries, variational principles, conservation laws
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 1 , January 2017 , pp. 93 - 125
Copyright
Copyright © Global-Science Press 2016 

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References

[1] Drazin, P.G., Solitons, in: London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1983.CrossRefGoogle Scholar
[2] Debussche, A. and Menza, L.D., Numerical simulation of focusing stochastic nonlinear Schrödinger equations, Physica D, 162 (2002), 131–154.Google Scholar
[3] Elgin, J.N., Stochastic perturbations of optical solitons, Opt. Lett., 18 (1993), 10–12.Google Scholar
[4] Gordon, J.P., Interacion forces among solitons in optical fibers, Opt. Lett., 8 (1983), 596–598.CrossRefGoogle Scholar
[5] Hasegawa, A., Optical solitons in fibers, Springer Berlin Heidelberg, 1989.Google Scholar
[6] Hairer, E., Lubich, C. and Wanner, G.. Geometric numerical integration, New York: Springer-Verlag, 2006.Google Scholar
[7] Jiang, S., Wang, L. and Hong, J., Stochastic multi-symplectic integrator for stochastic nonlinear Schrödinger equation, Commun. Comput. Phys., 14 (2013), 393–411.Google Scholar
[8] Kodama, Y., Romagnoli, M. and Wabnitz, S., Soliton stability and interactions in fibre lasers, Electron. Lett., 28 (1992), 1981–1983.Google Scholar
[9] Kong, L., Hong, J., Ji, L. and Zhu, P., Compact and efficient conservative schemes for coupled nonlinear Schrödinger equations, Numer. Methods Partial Differential Eq., 31 (2015), 1814–1843.Google Scholar
[10] Liu, W., Pan, N., Huang, L. and Lei, M., Soliton interactions for coupled nonlinear Schrödinger equations with symbolic computation, Nonlinear Dynam., 78 (2014), 755–770.CrossRefGoogle Scholar
[11] Lele, S.K., Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103 (1992), 16–42.CrossRefGoogle Scholar
[12] Mitschke, F.M. and Mollenauer, L.F., Experimental observation of interaction forces between solitons in optical fibers, Opt. Lett., 12 (1987), 355–357.Google Scholar
[13] Malomed, B.A., Bound solitons in coupled nonlinear Schrödinger equations, Phys. Rev. A, 45 (1992), R8321.CrossRefGoogle ScholarPubMed
[14] Sun, J., Gu, X. and Ma, Z., Numerical study of the soliton waves of the coupled nonlinear Schrödinger system, Physica D, 196 (2004), 311–328.Google Scholar
[15] Sun, Z., Gao, Y., Yu, X. and Liu, Y., Switching of bound vector solitons for the coupled nonlinear Schrödinger equations with nonhomogenously stochastic perturbations, Chaos, 22 (2012), 043132.Google Scholar
[16] Ueda, T. and Kath, W.L., Dynamics of optical pulses in randomly birefringent fibers, Physica D, 55 (1992), 166–181.Google Scholar
[17] Wai, P.K.A., Menyuk, C.R. and Chen, H., Stability of solitons in randomly varying birefringent fibers, Opt. Lett., 16 (1991), 1231–1233.CrossRefGoogle ScholarPubMed