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In this paper, we analyze the dynamics of a generalized Rotation-Camassa-Holmequation, 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 theRotation-Degasperis-Procesi (DP) equation for $\theta=\frac{1}{4}$). Our studyis conducted via the bifurcation method and qualitative theory of dynamicalsystems. The existence of not only smooth solitary wave solutions, periodicwave solutions, but also peakons and periodic peakon solutions is shown. Thechosen values $\theta$, allow us to assess the difference in behavior betweenthe classical $\theta$-equation and the Rotation-$\theta$ equation. We concludethat the Coriolis effect does affect the traveling wave solutions. We summarizethe bifurcations and explicit expressions of waves solutions in three theoremsin Section 4. A conclusion ends the paper.

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