×

zbMATH — the first resource for mathematics

Symplectic and multi-symplectic methods for the nonlinear Schrödinger equation. (English) Zbl 1050.65127
Summary: The Hamiltonian and the multi-symplectic formulations of the nonlinear Schrödinger equation are considered. For the multi-symplectic formulation, a new six-point difference scheme which is equivalent to the multi-symplectic Preissman integrator is derived. Numerical experiments are also reported.

MSC:
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Hasegawa, A., Optical solitons in fibers, (1989), Springer-Verlag Berlin
[2] Lamb, G.L., Elements of soliton theory, (1980), John Wiley & Sons New York · Zbl 0445.35001
[3] Holloway, J.P., On numerical methods for Hamiltonian PDE’s and a collocation method for the Vlasov-Maxwell equations, J. comput. phys., 129, 121-133, (1996) · Zbl 0862.65049
[4] Qin, M.Z.; Zhu, W.J., Construction of symplectic schemes for wave equations via hyperbolic functions sinh(x), cosh(x), and tanh(x), Computers math. applic., 26, 8, 1-11, (1993) · Zbl 0787.65061
[5] Tang, Y.F.; Vázquez, L.; Zhang, F.; Pérez-García, V.M., Symplectic methods for the nonlinear Schrödinger equation, Computers math. applic., 32, 5, 73-83, (1996) · Zbl 0858.65124
[6] Bridges, T.J., Multi-symplectic structures and wave propagation, Math. proc. Cam. phil. soc., 121, 147-190, (1997) · Zbl 0892.35123
[7] T.J. Bridges and S. Reich, Multi-symplectic integrators: Numerical schemes for Hamiltonian PDEs that conserve symplecticity, Technical Report. · Zbl 0984.37104
[8] Reich, S., Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations, J. comput. phys., 157, 473-499, (2000) · Zbl 0946.65132
[9] Schober, C.M., Symplectic integrators for Ablowitz-Ladik discrete nonlinear Schrödinger equation, Phys. lett. A, 259, 140-151, (1999) · Zbl 0935.37053
[10] Tang, Y.F.; Pérez-García, V.M.; Vázquez, L., Symplectic methods for the Ablowitz-Ladik model, Appl. math. comput., 82, 17-38, (1997) · Zbl 0870.65115
[11] Channell, P.J.; Scovel, J.C., Symplectic integration of Hamiltonian systems, Nonlinearity, 3, 2, 231-259, (1990) · Zbl 0704.65052
[12] Feng, K.; Qin, M.Z., (), 1-37
[13] Sanz-Serna, J.M.; Calvo, M.P., Numerical Hamiltonian problems, (1994), Chapman & Hall London · Zbl 0816.65042
[14] Richtmyer, R.D.; Morton, K.W., ()
[15] Feng, B.F.; Mitsui, T., A finite difference method for the Korteweg-de Vries and the Kadomtsev-Petviashvili equations, J. comput. appl. math., 90, 95-116, (1998) · Zbl 0907.65085
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.