Bifurcation and Exact Solutions of a Concatenation Physical Model

To find the exact explicit solution of the concatenation model, the corresponding differential system of the amplitude component is used, which is a planar dynamical system with a singular straight line. In this paper, by using the techniques from dynamical systems and singular traveling wave theory developed by Li and Chen [2007], its corresponding traveling wave system is solved and analyzed, obtaining the corresponding phase portraits and showing the dynamical behavior of the amplitude component. Under different parameter conditions, exact explicit solitary wave solutions, periodic wave solutions, kink and anti-kink wave solutions, compacton solutions, as well as peakons and periodic peakons, are found explicitly.