Construction Of Multi-Symplectic Scheme For "good" Boussinesq Equation

In this paper, The Hamiltonian formulations of the linear ood?Boussinesq equation and the mtilti-symplectic formulations of the nonlinear ood?Boussinesq equation are considered. For the multi-symplectic formulations, two new fifteenpoint difference schemes which are equivalent to the multi-symplectic Preissman integrator are derived. We use the hyperbolic functions tanh(x), sinh(x) and cosh(x) to construct symplectic schemes of arbitrary order for the linear ood?Boussinesq equation with periodic boundary condition. And show that the discretizations are rnulti-symplectic of arbitrary order accuracy for the modified linear ~good?Boussinesq equation when the periodic boundary condition vanishes. We also present numerical experiments, which show that the symplectic and multi-symplectic schemes have excellent long-time numerical behavior.The paper was supported by State Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Cornputing, Chinese Academy of Sciences.