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Numerical simulation of a nonlinearly coupled Schrödinger system: A linearly uncoupled finite difference scheme. (English) Zbl 1202.65116
A system of strong coupled 1D nonlinear Schrödinger equations in an arbitrary large limited in space and time domain at appropriate initial-boundary value conditions is considered. For solving the problem a conservative finite difference scheme is constructed, based on a regular three-level nine-point stencil. The scheme is uncoupled in computation for the two unknown discrete functions. It is shown that the difference scheme conserves the discrete energy and discrete masses in conformity with the corresponding properties of the differential problem. Using the energy method unconditional stability and second-order convergence of the scheme in the \(L_2\)-norm are proved.
Numerical tests are given to demonstrate advantages of the proposed scheme in comparison with the two-level conservative scheme obtained by W. J. Sonnier and C. I. Christov [Math. Comput. Simul. 69, No. 5–6, 514–525 (2005; Zbl 1119.65396)] and the multi-symplectic scheme by J.-Q. Sun and M.-Z. Qin [Comput. Phys. Commun. 155, No. 3, 221–235 (2003; Zbl 1196.65195)].

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q51 Soliton equations
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[1] Ablowitz, M.J., Solitons and the inverse transform, (1981), SIAM Philadelpha · Zbl 0463.35072
[2] Chang, Q.; Jiang, H., A conservative difference scheme for the Zakharov equations, J. comput. phys., 113, 309-319, (1994) · Zbl 0807.76050
[3] Chang, Q.; Guo, B.; Jiang, H., Finite difference method for the generalized Zakharov equations, Math. comput., 64, 537-553, (1995) · Zbl 0827.65138
[4] Christov, C.I.; Dost, S.; Maugin, G.A., Inelasticity of soliton collisions in system of coupled NLS equations, Phys. scripta, 50, 449-454, (1994)
[5] Furihata, D., Finite difference schemes for \((\partial u / \partial t) = (\partial / \partial x)^\alpha(\delta G / \delta u)\) that inherit energy conservation or dissipation property, J. comput. phys., 156, 181-205, (1999) · Zbl 0945.65103
[6] Furihata, D., Finite-difference schemes for nonlinear wave equation that inherit energy conservation property, J. comput. appl. math., 134, 37-57, (2001) · Zbl 0989.65099
[7] Gross, E.P., Hydrodynamics of a superfluid condensate, J. math. phys., 4, 195-207, (1963)
[8] Gross, E.P., Structure of a quantized vortex in boson systems, Il nuovo cimiento, 20, 454, (1961) · Zbl 0100.42403
[9] Hasegawa, A., Optical solitons in fibers, (1989), Springer-Verlag Berlin
[10] Ismail, M.S.; Alamri, S.Z., Highly accurate finite difference method for coupled nonlinear Schrödinger equation, Int. J. comput. math., 81, 333-351, (2004) · Zbl 1058.65090
[11] Ismail, M.S.; Taha, T.R., Numerical simulation of coupled nonlinear Schrödinger equation, Math. comput. simul., 56, 547-562, (2001) · Zbl 0972.78022
[12] Ismail, M.S.; Taha, T.R., A linearly implicit conservative scheme for the coupled nonlinear Schrödinger equation, Math. comput. simul., 74, 302-311, (2007) · Zbl 1112.65079
[13] Lamb, G.L., Elements of soliton theory, (1980), John Wiley Sons New York · Zbl 0445.35001
[14] Li, S.; Vu-Quoc, L., Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation, SIAM J. numer. anal., 32, 1839-1875, (1995) · Zbl 0847.65062
[15] T. Matsuo, New conservative schemes with discrete variational derivatives for nonlinear wave equations, J. Comput. Appl. Math. doi: 10.1016/j.cam.2006.03.009. · Zbl 1120.65096
[16] Matsuo, T.; Furihata, D., Dissipative or conservative finite difference schemes for complex-valued nonlinear partial differential equations, J. comput. phys., 171, 425-447, (2001) · Zbl 0993.65098
[17] Pitaevskii, L.P., Vortex lines in an imperfect Bose gas, Soviet phys. JETP, 13, 451-454, (1961)
[18] Sonnier, W.J.; Christov, C.I., Strong coupling of Schrödinger equations: conservative scheme approach, Math. comput. simul., 69, 514-525, (2005) · Zbl 1119.65396
[19] Sun, J.Q.; Qin, M.Z., Multi-symplectic methods for the coupled 1D nonlinear Schrödinger system, Comput. phys. commun., 155, 221-235, (2003) · Zbl 1196.65195
[20] Sun, J.Q.; Gu, X.Y.; Qin, M.Z., Multisymplectic difference schemes for coupled nonlinear Schrödinger system, Chin. J. comput. phys., 21, 321-328, (2004)
[21] Taha, T.R.; Ablovitz, M.J., Analytical and numerical aspects of certain nonlinear evolution equations. II. numerical, Schrödinger equation, J. comput. phys., 55, 203-230, (1984) · Zbl 0541.65082
[22] Wang, T.C.; Chen, J.; Zhang, L.M., Conservative difference methods for the klein – gordon – zakharov equations, J. comput. appl. math., 205, 430-452, (2007) · Zbl 1123.65091
[23] Wang, T.C.; Zhang, L.M., Analysis of some new conservative schemes for nonlinear Schrödinger equation with wave operator, Appl. math. comput., 182, 1780-1794, (2006) · Zbl 1161.65349
[24] Yang, J., Multi solitons perturbation theory for themanakov equations and its applications to nonlinear optics, Phys. rev. E, 59, 2393, (1999)
[25] Zhang, L.M., Convergence of a conservative difference scheme for a class of klein – gordon – schrödinger equations in one space dimension, Appl. math. comput., 163, 343-355, (2005) · Zbl 1080.65084
[26] Zhang, F., Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme, Appl. math. comput., 71, 165-177, (1995) · Zbl 0832.65136
[27] Zhang, L.M., A conservative numerical scheme for a class of nonlinear Schrödinger equation with wave operator, Appl. math. comput., 145, 603-612, (2003) · Zbl 1037.65092
[28] Zhou, Y.L., Application of discrete functional analysis to the finite difference methods, (1990), International Academic Publishers Beijing
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