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 oscilations 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 behavior of these solutions.