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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. [ABSTRACT FROM AUTHOR]

The bifurcation method of dynamical system is applied to study traveling waves of the Joseph—Egri equation. The phase space geometry of traveling wave system of the Joseph—Egri equation is investigated in detail. We obtain the parameter bifurcation sets in which various bounded and unbounded orbits are identified and simulated. Furthermore, by the calculation of complicated elliptic integrals, exact expressions of all traveling wave solutions of the Joseph—Egri equation are given, including bounded and unbounded ones. [ABSTRACT FROM AUTHOR]