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 analyse 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.

Keywords

Bibliographic record

BibTeX Citation

@article{Dutykh2016travellingwave,
  author = {Dutykh, D. and Ionescu--Kruse, D.},
  title = {Travelling wave solutions for some two-component shallow water models},
  journal = {J. Diff. Equations},
  year = {2016},
  volume = {261},
  number = {2},
  pages = {1099--1114},
  doi = {10.1016/j.jde.2016.03.035},
  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 analyse 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.},
  keywords = {travelling waves, cnoidal waves, solitary waves, phase-plane analysis, Camassa-Holm equations, Green-Naghdi model, Serre equations}
}