Gap-solitons in a three-layered stratified flow

In this paper, we give an explicit asymptotic construction of a class of solitary waves, widely known as gap-solitons in other physical contexts, for a certain three-layered fluid flow. The essential ingredients are the existence of a spectral gap between two branches of the dispersion relation, and the development of a set of envelope equations to describe weakly nonlinear waves, whose carrier frequency and wavenumber belong to the centre of this gap. Here we describe the gap-soliton solutions to this set of envelope equations. For the special case of particular interest when the envelope and carrier speeds are identical, so that the gap-soliton is a steady travelling wave of the full fluid system, we show that there is large class of such gap-solitons.