Dynamical behavior of traveling waves in a generalized VP-mVP equation with non-homogeneous power law nonlinearity

In this paper, we investigate the dynamical behavior of traveling waves for a generalized Vakhnenko-Parkes-modified Vakhnenko-Parkes (VP-mVP) equation with non-homogeneous power law nonlinearity. By the dynamical systems approach and the singular traveling wave theory, the existence of all possible bounded traveling wave solutions is discussed, including smooth solutions (solitary wave solutions, periodic wave solutions and breaking wave solutions) and non-smooth solutions (solitary cusp wave solutions and periodic cusp wave solutions). We not only obtain all the explicit parametric conditions for the existence of 5 kinds of bounded traveling wave solutions, but also give their exact explicit expressions. Moreover, we qualitatively analyze the dynamical behavior of these traveling waves by using the bifurcation of phase portraits under different parameter conditions, and strictly prove the evolution of different traveling waves with their exact expressions.