  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.
