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Multi-symplectic methods for the coupled 1D nonlinear Schrödinger system. (English) Zbl 1196.65195
Summary: The multi-symplectic formulation of the coupled 1D nonlinear Schrödinger system (CNLS) is considered. For the multi-symplectic formulation, a new six point scheme, which is equivalent to the multi-symplectic Preissman integrator, is derived. We also present numerical experiments, which show that the multi-symplectic scheme has excellent long-time numerical behaviour and energy conservation property.

MSC:
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
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[1] Hasegawa, A., Optical solitons in fibers, (1989), Springer-Verlag Berlin
[2] Bridge, T.J., Multi-symplectic structures and wave propagation, Math. proc. Cambridge philos. soc., 121, 147-190, (1997) · Zbl 0892.35123
[3] Marsden, J.E.; Shkoller, S., Multisymplectic geometry, covariant Hamiltonians and water waves, Math. proc. Cambridge philos. soc., 125, (1999) · Zbl 0922.58029
[4] Feng, B.F.; Mitsui, T., A finite difference method for the kortewegde Vries and the kadomtsev – petviashvili equations, J. comput. appl. math., 90, 95-116, (1998) · Zbl 0907.65085
[5] Bridges, T.J.; Reich, S., Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplectic, Phys. lett. A, 284, 184-193, (2001) · Zbl 0984.37104
[6] Liu, T.T.; Qin, M.Z., Multisymplectic geometry and multi-symplectic preissman scheme for the KP equation, Jmp, 43, 8, 4060-4077, (2002) · Zbl 1060.37050
[7] Feng, K.; Qin, M.Z., The symplectic methods for computation of Hamiltonian equations, (), 1-37
[8] Qin, M.Z.; Zhu, W.J., Construction of symplectic schemes for wave equation via hyperbolic function sinh(x), cosh(x) and tanh(x), Comput. math. appl., 26, 8, 1-11, (1993)
[9] Bridges, T.J., Multisymplectic structures and wave propagation, Math. proc. Cambridge philos. soc., 121, 147-190, (1997) · Zbl 0892.35123
[10] Reich, S., Multi-symplectic runge – kutta methods for Hamiltonian wave equations, Jcp, 157, 473-499, (2000) · Zbl 0946.65132
[11] Yang, J., Multi solitons perturbation theory for the Manakov equations and its applications to nonlinear optics, Phys. rev. E, 59, 2, 2393, (1999)
[12] Ablowitz, M.J.; Segur, H., Solitions and the inverse scattering transform, (1981), SIAM Philadelphia · Zbl 0299.35076
[13] Zhao, P.F.; Qin, M.Z., Multisymplectic geometry and multi-symplectic preissman scheme for the KdV equation, J. phys. A math. geom., 33, 3613-3626, (2000) · Zbl 0989.37062
[14] Wang, Y.S.; Qin, M.Z., Multisymplectic geometry and multisymplectic schemes for the nonlinear klein – gordon equation, J. phys. soc. Japan, 70, 3, 653-661, (2001) · Zbl 1113.37308
[15] Sun, Y.J.; Qin, M.Z., Construction of multi-symplectic scheme of any finite order for modified wave equation, Jmp, 41, 11, 7854-7868, (2000) · Zbl 0992.65145
[16] Chen, J.B.; Qin, M.Z.; Tang, Y.F., Symplectic and multi-symplectic method for the nonlinear Schrödinger equation, Comput. math. appl., 43, 1095-1106, (2002) · Zbl 1050.65127
[17] Zhang, W.P.; Huang, L.Y.; Qin, M.Z., The multi-symplectic algorithm for “good” Boussinesq equation, Appl. math. mech., 23, 7, 835-841, (2002) · Zbl 1013.76063
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