Effect of the Coriolis force on bounded traveling waves of the rotation-two-component Camassa–Holm system

Abstract

In this paper, we qualitatively study the effect of the Coriolis force on traveling waves of the rotation-two-component Camassa–Holm system in the rotating fluid, which is a model in the equatorial water waves, from the perspective of dynamical systems. We obtain the explicit critical rotational speed $\mathrm{\Omega }=\frac{1}{2(A+c)}$ (A is a parameter of the system characterizing a linear underlying shear flow and c is the wave speed), which reflects the Coriolis force. Based on this critical vorticity, we obtain phase portraits of the system under exact explicit parameters conditions. Then we not only show the existence of various bounded traveling waves including smooth solitary waves, peakons and kink waves, under corresponding exact explicit parameters conditions, but also obtain their exact expressions. Of particular interest is that we find that kink waves (which were not found previouly) exist only when $\mathrm{\Omega }>\frac{1}{2(A+c)}$, that is, the rotational speed Ω should be greater than $\frac{1}{2(A+c)}$.

Keywords:

• Rotating two-component Camassa–Holm system
• Coriolis force
• solitary waves
• peakons
• kink waves

MSC (2010):

• 34C60
• 35B32
• 35C07

Acknowledgements

This research is partially supported by the National Natural Science Foundation of China (12071162), and the Fundamental Research Funds for the Central Universities (No. ZQN-802).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research is partially supported by the National Natural Science Foundation of China (12071162), and the Fundamental Research Funds for the Central Universities (No. ZQN-802).