Bifurcations and exact traveling wave solutions for a highly nonlinear shallow water wave model

Under certain conditions on parameters, the corresponding traveling wave system of a highly nonlinear shallow water wave model is a planar dynamical system which has two singular straight lines. Corresponding dynamical system is detailed by utilizing dynamical system methods and singular traveling wave theory techniques. Additionally, this work gives bifurcations of phase portraits. Then, the dynamical behavior is derived. Under some specified parameter conditions, exact explicit solitary wave solutions, periodic wave solutions, peakons, periodic peakons and compacton solutions can be found.