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

Considered here is a Boussinesq system of equations from surface water wave theory. The particular system is one of a class of equations derived and analyzed in recent studies. After a brief review of theoretical aspects of this system, attention is turned to numerical methods for the approximation of its solutions with appropriate initial and boundary conditions. Because the system has a spatial structure somewhat like that of the Korteweg–de Vries equation, explicit schemes have unacceptable stability limitations. We instead implement a highly accurate, unconditionally stable scheme that features a Galerkin method with periodic splines to approximate the spatial structure and a two-stage Gauss–Legendre implicit Runge-Kutta method for the temporal discretization. After suitable testing of the numerical scheme, it is used to examine the travelling-wave solutions of the system. These are found to be generalized solitary waves, which are symmetric about their crest and which decay to small amplitude periodic structures as the spatial variable becomes large.