In this paper, we consider two types of traveling wave systems of the generalized Kundu-Mukherjee-Naskar equation. Firstly, due to the integrity, we obtain their energy functions. Then, the dynamical system method is applied to study bifurcation behaviours of the two types of traveling wave systems to obtain corresponding global phase portraits in different parameter bifurcation sets. According to them, every bounded and unbounded orbits can be identified clearly and investigated carefully which correspond to various traveling wave solutions of the generalized Kundu-Mukherjee-Naskar equation exactly. Finally, by integrating along these orbits and calculating some complicated elliptic integral, we obtain all type I and type II traveling wave solutions of the generalized Kundu-Mukherjee-Naskar equation without loss.
Review Article In this paper, the bifurcation theory of dynamical system is applied to study the traveling waves of the (3+1)dimensional Zakharov-Kuznetsov Equation with Power Law Nonlinearity. By transforming the traveling wave system of the Zakharov-Kuznetsov equation into a dynamical system in , we derive various parameter conditions which guarantee the existence of its bounded and unbounded orbits. Furthermore, by calculating complicated elliptic integrals along these orbits, we obtain exact expressions of bounded traveling wave solutions of the (3+1)-dimensional Zakharov-Kuznetsov equation for n=1.
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