Bifurcations and Exact Traveling Wave Solutions for the Model of Slightly Dispersive Quasi-Incompressible Hyperelastic Materials

This paper is devoted to the bifurcations and exact traveling wave solutions in a model of slightly dispersive quasi-incompressible hyperelastic materials by introducing a very general response function for the Cauchy stress tensor of dispersive hyperelastic solids, which was recently established by Saccomandi and Vergor [1]. By using the method of dynamical systems, we study the bifurcations of phase portraits and dynamical behaviors of solutions for the corresponding traveling wave system, which is a planar dynamical system. In addition to the results on the existence of solitary wave and kink wave solutions that has been proved, we not only find more traveling wave solutions, such as periodic wave and anti-kink wave solutions, but also obtain the exact explicit parametric representations and bifurcations of all possible traveling wave solutions under various parameter conditions.