Numerical solution of Boussinesq systems of KdV–KdV type: II. Evolution of radiating solitary waves

In this paper we consider a coupled KdV system of Boussinesq type and its symmetric version. These systems were previously shown to possess generalized solitary waves consisting of a solitary pulse that decays symmetrically to oscillations of small, constant amplitude. We solve numerically the periodic initial-value problem for these systems using a high order accurate, fully discrete, Galerkin-finite element method. (In the case of the symmetric system, it is possible to prove rigorous, optimal-order, error estimates for this scheme.) The numerical scheme is used in an exploratory fashion to study radiating solitary-wave solutions of these systems that consist, in their simplest form, of a main, solitary-wave-like pulse that decays asymmetrically to small-amplitude, outward-propagating, oscillatory wave trains (ripples). In particular, we study the generation of radiating solitary waves, the onset of ripple formation and various aspects of the interaction and long time behaviour of these solutions.