WAVE MOTION 11 (1989) 261-269 261 NORTH-HOLLAND SOLITONS IN A SYSTEM OF COUPLED KORTEWEG-DE VRIES EQUATIONS Yuri S. KIVSHAR Institute for Low Temperature Physics and Engineering, UkrSSR Academy of Sciences, 47 Lenin Avenue, Kharkov 310164, U.S.S.R. Boris A. MALOMED P.P. Shirshov Institute of Oceanology, 23 Krasikov Street, Moscow 117218, U.S.S.R.* Received 3 February 1988 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. 1. Introduction The present paper is the continuation of a preceding paper of the second author [1] devoted to the investigation of soliton dynamics in the framework of the Gear-Grimshaw model [2, 3]: Ult --6UlUlx + Ulxxx --- --elu2Uzx -- ez( UlU2)x -- eau2 .... (1) U2t -- 6flU2U2xq-/3U2xxx -- Vou2x = -or [8l(Ul u2) x -I- EzUlt/ix d- E3Ulxxx ]. (2) These equations describe a resonant interaction of two transverse internal gravity-wave modes in a shallow stratified liquid. In (1), (2), the functions Ul and u2 stand for wave variables in the two wave modes, el, e2 and e 3 are inter-mode coupling constants which in the present paper will be regarded as small parameters, while the parameters/3, a and Vo are arbitrary. As has been revealed in the numerical experiments by Gear and Grimshaw [2, 3], two solitons belonging to different wave modes undergo strong interaction provided that their relative velocity is sufficiently small. This interaction may result in forming a bound oscillatory state of the two solitons, so that their motion looks like "leapfrogging". A theoretical description of the "leapfrogging" two-soliton dynamics has been put forward in paper [1] on the basis of a perturbation theory for solitons (see, e.g., [4-7]). In * Address for correspondence. 0165-2125/89/$3.50 O 1989, Elsevier Science Publishers B.V. (North-Holland)