Solitary waves of a coupled Korteweg-de Vries system

In the long-wave, weakly nonlinear limit a generic model for the interaction of two waves with nearly coincident linear phase speeds is a pair of coupled Korteweg-de Vries equations. Here we consider the simplest case when the coupling occurs only through linear non-dispersive terms, and for this case delineate the various families of solitary waves that can be expected. Generically, we demonstrate that there will be three families, (a) pure solitary waves which decay to zero at in nity exponentially fast, (b) generalized solitary waves which may tend to small-amplitude oscillations at in nity, and (c) envelope solitary waves which at in nity consist of decaying oscillations. We use a combination of asymptotic methods and the rigorous results obtained from a normal form approach to determine these solitary wave families.