With The Wave Operator Of The Nonlinear Schr (?) Dinger Equation For Symplectic Scheme

We consider a class of nonlinear Schrodinger equation with wave operator in this paper. Because it has multi-symplectic structure, we turn it into mulit-symplectic Hamiltonian formulations by introducing canonical momenta, and find that it has the properties of conservation law of multi-symplecticity, local energy and local momentum. The discretization obtained by applying the method of Gauss-Legendre to the space direction of the formulations has multi-symplectic conservation law of semi-discretizaton ;furthermore, the multi-symplectic integrator is gained by concatenating the Guass-Legendre discretization of the time direction.We get the Preissman multi-symplectic integrator via implicit midpoint rule. At the same time, through eliminating variables, we can gain a new nine points multi-symplectic scheme, which is equivalent to the multi-symplectic Preissman integrator. We also construct semi-discrete and full-discrete schemes of the formulations through symplectic Fourier pseudospectral method. These schemes satisfy semi-discrete and full-discrete multi-symplectic conservation law respectively. The results of the experiments, which we do with nine points scheme and full-discrete Fourier pseudospectral multi-symplectic scheme , show that , the schemes we construct in this paper are effective with excellent long-time numerical behavior , and that the Fourier pseudospectral multi-symplecitc scheme has higher error accuracy than the new nine points scheme.