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.
Highlights
Water wave theory has been traditionally based on the irrotational flow assumption [18]
The modern and large-amplitude theory for periodic surface water waves with a general vorticity distribution was established by Constantin and Strauss in [16], an investigation which initiated an intense study of waves with vorticity — see, for example, [13, 17, 43] and the references therein
The choice of constant vorticity is not just a mathematical simplification since for waves propagating at the surface of water over a nearly flat bed, which are long compared to the mean water depth, the existence of a non-zero mean vorticity is more important than its specific distribution
Summary
Water wave theory has been traditionally based on the irrotational (and potential) flow assumption [18] It allows to determine all components of the velocity vector by taking partial derivatives of just one scalar function — the so-called velocity potential. In the approximate theories of long waves on flows with an arbitrary vorticity distribution, Freeman and Johnson [22] derived, by the use of asymptotic expansion, a KdV equation with the coefficients modified to include the effect of shear. In our previous study [20] we provided a complete phase plane analysis of all possible travelling wave solutions which may arise in several twocomponent systems which model the propagation of shallow water waves, namely, the Green–Naghdi equations, the integrable two-component Camassa–Holm equations and a new two-component system of Green–Naghdi type. Chen [7] found numerically multi-pulse travelling wave solutions to the (KB) system, solutions which consist of an arbitrary number of troughs
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
