Effects of vorticity on the travelling waves of some shallow water two-component systems

Abstract

In the present study we consider three two-component (integrable and non-integrable) systems which describe the propagation of shallow water waves on a constant shear current. Namely, we consider the two-component Camassa-Holm equations, the Zakharov-Ito system and the Kaup—Boussinesq equations all including constant vorticity effects. We analyze both solitary and periodic-type travelling waves using the simple and geometrically intuitive phase space analysis. We get the pulse-type solitary wave solutions and the front solitary wave solutions. For the Zakharov-Ito system we underline the occurrence of the pulse and anti-pulse solutions. The front wave solutions decay algebraically in the far field. For the Kaup-Boussinesq system, interesting analytical multi-pulsed travelling wave solutions are found.

Keywords

Bibliographic record

BibTeX Citation

@article{Dutykh2019effectsvorticity,
  author = {Dutykh, D. and Ionescu-Kruse, D.},
  title = {Effects of vorticity on the travelling waves of some shallow water two-component systems},
  journal = {Discrete & Continuous Dynamical Systems — A},
  year = {2019},
  volume = {39},
  number = {9},
  pages = {5521--5541},
  doi = {10.3934/dcds.2019225},
  abstract = {In the present study we consider three two-component (integrable and non-integrable) systems which describe the propagation of shallow water waves on a constant shear current. Namely, we consider the two-component Camassa-Holm equations, the Zakharov-Ito system and the Kaup—Boussinesq equations all including constant vorticity effects. We analyze both solitary and periodic-type travelling waves using the simple and geometrically intuitive phase space analysis. We get the pulse-type solitary wave solutions and the front solitary wave solutions. For the Zakharov-Ito system we underline the occurrence of the pulse and anti-pulse solutions. The front wave solutions decay algebraically in the far field. For the Kaup-Boussinesq system, interesting analytical multi-pulsed travelling wave solutions are found.},
  keywords = {two-component systems, solitary waves, vorticity effects, cnoidal waves, analytical solutions, waves on shear flow, phase-plane analysis, periodic waves, multi-pulsed solutions, shallow water waves, phase space analysis, anti-pulse solutions, Kaup-Boussinesq equations, pulse-type solutions, Camassa-Holm equations, front wave solutions, Zakharov-Ito system}
}