Traveling waves of a generalized Rotation-Camassa-Holm equation with the Coriolis effect

TL;DR

This paper investigates the effects of Earth's rotation on a generalized family of wave equations, revealing how the Coriolis effect influences various traveling wave solutions including smooth, peakon, and periodic waves.

Contribution

It introduces the Rotation-$ heta$ equation, extending classical models to include Coriolis effects and analyzes their wave solutions using bifurcation and dynamical systems methods.

Findings

Existence of smooth, peakon, and periodic wave solutions.

Coriolis effect influences the shape and bifurcation of traveling waves.

Explicit wave solutions and bifurcation scenarios are characterized.

Abstract

In this paper, we analyze the dynamics of a generalized Rotation-Camassa-Holm equation, which is the $\theta$-equation augmented with the Coriolis effect, induced by the earth rotation. The generalized Rotation-Camassa-Holm equation (named as Rotation-$\theta$ equation) is a generalization of a family of models (including the Rotation-Camassa-Holm equation for $\theta=\frac{1}{3}$, asymptotic Rotation-Camassa-Holm equation for $\theta=1$ and the Rotation-Degasperis-Procesi (DP) equation for $\theta=\frac{1}{4}$). Our study is conducted via the bifurcation method and qualitative theory of dynamical systems. The existence of not only smooth solitary wave solutions, periodic wave solutions, but also peakons and periodic peakon solutions is shown. The chosen values $\theta$, allow us to assess the difference in behavior between the classical $\theta$-equation and the Rotation-$\theta$ equation. We conclude that the Coriolis effect does affect the traveling wave solutions. We summarize the bifurcations and explicit expressions of waves solutions in three theorems in Section 4. A conclusion ends the paper.