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

  • * Corresponding author: Delia Ionescu-Kruse

    * Corresponding author: Delia Ionescu-Kruse
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  • 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.

    Mathematics Subject Classification: 74J30, 76F10, 35C07, 76B25, 70K05, 35Q35.


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  • Figure 1.  Sketch of the fluid domain: free surface flow on a shear current.

    Figure 2.  The graph of the inequality $c \mathfrak{c}^{+} > 1 $ against the vorticity Ω.

    Figure 3.  The phase-portrait and the solitary wave profiles in the CH2 model with constant vorticity, for: (a) $ c \mathfrak{c}^{+}>0$; (b) $ c \mathfrak{c}^{+} = 0$. We highlight the fact that two fronts tend only algebraically to the equilibrium state H = 1.

    Figure 4.  The phase-portrait and the solitary wave profiles in the ZI model with constant vorticity, for: (a) $ c \mathfrak{c}^{+}>1$; (b) $c \mathfrak{c}^{+} = 1 $. We highlight the fact that two fronts tend only algebraically to the equilibrium state H = 1.

    Figure 5.  The phase-portrait and the wave profiles for the ZI model with constant vorticity, in the case the polynomial $ \mathcal{R}(H)$ has only two real roots, H1 < 0 and H2 > 0 and the constant $ \mathcal{K}_{1}<0$.

    Figure 6.  The phase-portrait and the wave profiles for the ZI model with constant vorticity in the case the polynomial $ \mathcal{R}(H)$ has four real roots: H1 < 0 and 0 < H2 < H3 < H4 , and the constant: (a) $ \mathcal{K}_{1}﹥0$; (b) $ \mathcal{K}_{1}<0$.

    Figure 7.  The phase-portrait and the wave profiles for the ZI model with constant vorticity in the case the polynomial $ \mathcal{R}(H)$ has four real roots: H1 < H2< H3 < 0 and H4 > 0, and the constant: (a) $ \mathcal{K}_{1}﹥0$; (b) $ \mathcal{K}_{1}<0$.

    Figure 8.  Analytical expressions (36) and (35), respectively, for different values of the constant vorticityand of the speed of propagation c.

    Figure 9.  Multi-pulse travelling wave solutions with two troughs.

    Figure 10.  One-trough travelling wave solutions to the KB system based on analytical formulas (42) and (43).

    Figure 11.  Analytical expressions (49) and (47), respectively, for different values of the constant vorticityand of the speed of propagation c.

    Figure 12.  The periodic velocity profile for the KB system in the case the polynomial $ \mathcal{P}(H)$ has one positive real root, one negative real root and two complex conjugate roots.

    Figure 13.  The periodic velocity profiles for the KB system in the case the polynomial $ \mathcal{P}(H)$ has two positive real roots and two negative real roots.

    Figure 14.  The periodic velocity profiles for the KB system in the case the polynomial $ \mathcal{P}(H)$ has three real positive roots and one negative real root.

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