×

zbMATH — the first resource for mathematics

A fourth-order explicit schemes for the coupled nonlinear Schrödinger equation. (English) Zbl 1133.65063
Summary: We derive numerical methods for solving the coupled nonlinear Schrödinger equation. We discretize the space derivative by central difference formulas of fourth-order. The resulting ordinary differential system is solved by the fourth-order explicit Runge-Kutta method. Neumann and periodic boundary conditions are used. The method is tested for accuracy and the conserved quantities. These methods conserve the three conserved quantities exactly for at least five decimal places. A comparison has been made with some existing methods.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Griffiths, D.F.; Mitchell, A.R.; Li Morris, J., A numerical study of the nonlinear Schrödinger equation, Comput. meth. appl. mech. eng., 45, 177-215, (1984) · Zbl 0555.65060
[2] Ismail, M.S.; Alamri, S.Z., Highly accurate finite difference method for coupled nonlinear Schrödinger equation, Int. J. comp. math., 81, 3, 333-351, (2004) · Zbl 1058.65090
[3] Ismail, M.S.; Taha, Thiab R., Numerical simulation of coupled nonlinear Schrödinger equation, Math. comp. simul., 56, 547-562, (2001) · Zbl 0972.78022
[4] M.S. Ismail, Thiab R Taha, A linearly implicit conservative scheme for the coupled nonlinear Schrödinger equation, Math. Comp. Simul., in press. · Zbl 1112.65079
[5] Sanz-Serna, J.M.; Verwer, J.G., Conservative and nonconservative Schrödinger equation, IMA J. numer. anal., 6, 25-42, (1986) · Zbl 0593.65087
[6] Shamerdan, A.B., The numerical treatment of the nonlinear schr ödinger equation, Comput. math. appl., 19, 67-73, (1990)
[7] Sheng, Q.; Khaliq, A.Q.M.; al- Said, E.A., Solving the generalized nonlinear Schrödinger equation via quartic spline approximation, J. comput. phys., 166, 400-417, (2001) · Zbl 0979.65082
[8] Sun, J.Q.; Quin, M.Z., Multisymplectic methods for the coupled 1D nonlinear Schrödinger system, Comp. phys. commun., 155, 221-235, (2003)
[9] Sun, J.Q.; Gu, X.Y.; Q Ma, Z., Numerical study of the soliton waves of the coupled nonlinear Schrödinger system, Physica D, 196, 311-328, (2004) · Zbl 1056.65083
[10] Wadati, M.; Izuka, T.; Hisakado, M., A coupled nonlinear Schrödinger equation and optical solitons, J. phys. soc. jpn., 61, 7, 2241-2245, (1992)
[11] Xu, Z.; He, J.; Han, H., Semi implicit operator splitting pade’ method for higher-order nonlinear Schrödinger equations, Appl. math. comput., 179, 596-605, (2006) · Zbl 1102.65093
[12] Zhang, F.; Perez_Garacia, V.M.; Vazquez, L., Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme, Appl. math. comput., 71, 164-177, (1995)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.