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Symplectic and multi-symplectic methods for coupled nonlinear Schrödinger equations with periodic solutions. (English) Zbl 1196.65139
Summary: We consider for the integration of coupled nonlinear Schrödinger equations with periodic plane wave solutions a splitting method from the class of symplectic integrators and the multi-symplectic six-point scheme which is equivalent to the Preissman scheme. The numerical experiments show that both methods preserve very well the mass, energy and momentum in long-time evolution. The local errors in the energy are computed according to the discretizations in time and space for both methods. Due to its local nature, the multi-symplectic six-point scheme preserves the local invariants more accurately than the symplectic splitting method, but the global errors for conservation laws are almost the same.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
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