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

Denys Dutykh; Delia Ionescu-Kruse

doi:10.3934/dcds.2019225

Article Contents

2019, Volume 39, Issue 9: 5521-5541. Doi: 10.3934/dcds.2019225

This issue Previous Article Topologies of continuity for Carathéodory delay differential equations with applications in non-autonomous dynamics Next Article

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

Denys Dutykh^1,2, and
Delia Ionescu-Kruse^3, ,

1.
Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France
2.
LAMA, UMR 5127 CNRS, Université Savoie Mont Blanc, Campus Scientifique, 73376 Le Bourget-du-Lac Cedex, France
3.
Simion Stoilow Institute of Mathematics of the Romanian Academy, Research Unit No. 6, P.O. Box 1-764, 014700 Bucharest, Romania

^* Corresponding author: Delia Ionescu-Kruse
^* Corresponding author: Delia Ionescu-Kruse

Received: December 31, 2018

Revised: March 31, 2019

Published: May 2019

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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:

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.

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

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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.

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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.

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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$.

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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$.

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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$.

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

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

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

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

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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.

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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.

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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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