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.
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Sketch of the fluid domain: free surface flow on a shear current.
The graph of the inequality
The phase-portrait and the solitary wave profiles in the CH2 model with constant vorticity, for: (a)
The phase-portrait and the solitary wave profiles in the ZI model with constant vorticity, for: (a)
The phase-portrait and the wave profiles for the ZI model with constant vorticity, in the case the polynomial
The phase-portrait and the wave profiles for the ZI model with constant vorticity in the case the polynomial
The phase-portrait and the wave profiles for the ZI model with constant vorticity in the case the polynomial
Analytical expressions (36) and (35), respectively, for different values of the constant vorticity Ω and of the speed of propagation c.
Multi-pulse travelling wave solutions with two troughs.
One-trough travelling wave solutions to the KB system based on analytical formulas (42) and (43).
Analytical expressions (49) and (47), respectively, for different values of the constant vorticity Ω and of the speed of propagation c.
The periodic velocity profile for the KB system in the case the polynomial
The periodic velocity profiles for the KB system in the case the polynomial
The periodic velocity profiles for the KB system in the case the polynomial