Solitons in a System of Coupled Korteweg-de Vries Equations Boris A. Malomed

A system of two Korteweg-de Vries equations coupled by small linear and nonlinear terms, which is a model of a resonant interaction between two internal gravity-wave modes in a shallow stratified liquid, is considered. The present paper is a continuation of a preceding one, where only the linear coupling was dealt with. Various dynamical processes involving one-mode solitons are investigated. It is demonstrated that two solitons belonging to the different wave modes may form an oscillatory bound state (a bi-soliton) which provides an explanation for the numerical results of Gear & Grimshaw demonstrating leapfrogging motion of the two interacting solitons. In the framework of a perturbation theory based on the inverse scattering transform, the frequency of the bi-soliton's internal oscillations is found, and emission of radiation by a weakly excited bi-soliton is studied. Phase shifts and radiative energy losses accompanying a collision between two free solitons belonging to the different modes are calculated. In addition, a collision between a free soliton and a bi-soliton is considered, and it is demonstrated that the collision may result in break-up of the bi-soliton.