Travelling wave solutions for some two-component shallow water models

Abstract

In the present study we perform a unified analysis of travelling wave solutions to three different two-component systems which appear in shallow water theory. Namely, we analyze the celebrated Green–Naghdi equations, the integrable two-component Camassa–Holm equations and a new two-component system of Green–Naghdi type. In particular, we are interested in solitary and cnoidal-type solutions, as two most important classes of travelling waves that we encounter in applications. We provide a complete phase-plane analysis of all possible travelling wave solutions which may arise in these models. In particular, we show the existence of new type of solutions.