Bifurcations and Exact Solutions in a Model of Hydrogen-Bonded-Chains

A model of dynamics of protons in hydrogen-bonded quasi-one-dimensional networks was derived, which is a singular system of the second kind with three parameters and two singular straight lines. In this paper, we use the method of dynamical systems to discuss the bifurcations of phase portraits of the vector fields defined by the singular system. Corresponding to the phase orbits of the system in different parameter conditions, we compute all possible exact parametric representations of solutions. It is shown that in given parameter conditions, there exist solitary wave solutions, kink wave solutions and periodic wave solutions. The mentioned system has no peakon solution.