The authors deal with the hydrogen-bonded system, that is \[ u_{tt}-u_{xx}-\beta u_xu_{xx}+\omega^2\frac{dV}{dt}=0,\tag{1} \] where \(V(u)=\frac{2(\cos(\frac u2)-\cos(\frac{u_0}{2}))^2}{1-\cos^2 (\frac{u_0}{2})}\), \(u_0\neq 0\) is a constant, and \(0\leq\cos\frac{u_0} {2}<1\), \(\beta,\omega\in\mathbb{R}\). The purpose of this work is to investigate all bifurcations of solitary wave, kink waves and periodic waves for (1). To this end the authors study all periodic annuli, homoclinic and heteroclinic orbits of \[ (c^2-1)\varphi_{\xi\xi}-\beta\varphi_\xi \varphi_{\xi\xi}-\frac{2\omega^2}{1-\cos^2(\frac{u_0}{2})}\sin\frac {\varphi}{2}\left(\cos\frac{\varphi}{2}-\cos\frac{u_0}{2}\right)=0\tag{2} \] depending on the parameter space of this system. Here \(u(x,t)= \varphi(x-ct)=\varphi(\xi)\).