Numerical solution of coupled KdV systems of Boussinesq equations: I. The numerical scheme and existence of generalized solitary waves

We consider some Boussinesq systems of water wave theory, which are of coupled KdV type. After a brief review of the theory of existence-uniqueness of solutions of the associated initial-value problems, we turn to the numerical solution of their initialand periodic boundary-value problems by unconditionally stable, highly accurate methods that use Galerkin/finite element type schemes with periodic splines for the spatial and the two-stage GaussLegendre implicit Runge-Kutta method for the temporal discretization. These systems are shown to possess generalized solitary wave solutions, wherein the main solitary wave pulse decays to small amplitude periodic solutions. Solutions of this type are constructed and studied by numerical means.