"Leapfrogging" solitons in a system of coupled KdV equations

A system of two KdV equations coupled by small linear dispersive terms is considered. This system describes, for example, resonant interaction of two transverse gravity internal wave modes in a shallow stratified liquid. In the framework of an approach based on Hamilton's equations of motion, evolution equations for parameters of two solitons belonging to different wave modes are obtained in the adiabatic approximation. It is demonstrated that when the solitons' velocities are sufficiently lose, the solitons may form a breather-like oscillatory bound state, which provides a natural explanation for recent numerical experiments demonstrating "leapfrogging" motion of the two solitons. The frequency and the maximum amplitude of the "breather"'s internal oscillations are obtained. For the case when the relative velocity of the solitons is not small, perturbation-induced phase shifts of the two colliding free solitons are calculated. Then emission of radiation (small-amplitude quasilinear waves) by an oscillating "breather," also detected in the numerical experiments, is investigated in the framework of the perturbation theory based on the inverse scattering transform. The intensity of the emission is calculated. Radiative effects accompanying collision of the free solitons are also investigated.