Efficient Multi-symplectic Algorithms Of The Coupled Nonlinear Schrodinger Equations

Hamiltonian system is widely used in natural process. All real physical process with negligible dissipation can be cast in suitable Hamiltonian formulation in phase space with symplectic structure, which means the symplectic-preserving algorithms can hold this properties naturally.Therefore, compared with general algorithms, symplectic and multi-symplectic algorithms, in some sense, have better performances of long-time numerical stability, fidelity of track, and preservation of conservation laws for Hamiltonian system. Due to their own constrains, the traditional symplectic and multi-symplectic algorithms usually have low efficiency and accuracy, which restrict their application to complex or higher dimensional system. Considering the coupled nonlinear Schr?dinger equations with multiply components, we propose two efficient multi-symplectic schemes: a semi-explicit multi-symplectic Fourier pseudospectral scheme and a semi-explicit multi-symplectic splitting scheme. The main contributions of this paper go as follows:1.We propose a semi-explicit multi-symplectic scheme to solve the 2-coupled nonlinear Schr?dinger equations. The scheme is derived by multi-symplectic Fourier pseudospectral method in spatial discretization and symplectic Euler method in temporal discretization. It is verified that the proposed multi-symplectic scheme has corresponding discrete multi-symplectic conservation laws. Numerical experiments show the good conservative properties of the proposed method during long-time numerical calculation.2.For 3-coupled nonlinear Schr?dinger equation, a semi-explicit multi-symplectic splitting scheme. Based on its multi-symplectic formulation,the 3-CNLS equation can be split into one linear multi-symplectic subsystem and one nonlinear infinite-dimensional Hamiltonian subsystem. For the linear subsystem, multi-symplectic Fourier pseudospectral method and symplectic Euler method are employed in spatial discretization and temporal discetization, respectively. For the nonlinear subsystem,the mid-point symplectic scheme is used. Numerical experiments also show effectiveness of the proposed method during long-time numerical calculation.