Numerical Methods Of Nonlinear Schr(?)dinger Equation

Finite difference methods and symplectic methods of nonlinear Schrodinger equation have been given out, and symplectic methods of nonlinear Schrodinger equation have been extended to higher dimension.Firstly, seven kinds of difference schemes of nonlinear Schrodinger equation have been established. They are two-order bi-level scheme, the alternating explicit-implicit scheme of the two-order bi-level scheme, four-order bi-level scheme, the alternating explicit-implicit scheme of the four-order bi-level scheme, two-order leap-frog scheme, four-order leap-frog scheme and two-order three levels scheme. The analyses of local truncation error of these seven kinds of difference schemes have been presented. The stability of these seven kinds of difference schemes have been analyze with frozen coefficient method, furthermore their convergence have been analyze. Their stability have been confirmed by the numerical experiments. Their truncation error and speeds have also been compared by numerical experiments.Then it has been proved that Euler midpoint schemes and leap-frog schemes of Hamiltonian system are symplectic schemes. Two-order Euler midpoint scheme, four-order Euler midpoint scheme, two-order leap-frog scheme and four-order leap-frog scheme of Hamiltonian system for nonlinear Schrodinger equation have been given out. And their numerical experiments have been made. Their feasibility have been confirmed by the numerical experiments. Then the results of two-order leap-frog scheme (one of the symplectic schemes) and two-order bi-level scheme (one of the non-symplectic schemes) have been compared by the numerical experiments.At last symplectic schemes of Hamiltonian system for nonlinear Schrodinger equation have been extended to higher dimension. Then two-order leap-frog schemes for nonlinear hyperbolic Schrodinger equation are given out and the numerical experiments are made to confirm it's feasibility.