Exact periodic wave solutions of the cubic-quintic Zakharov equation and their evolution with Hamilton energy

Abstract

In this paper, we study the exact periodic wave solutions of the Zakharov equation with cubic and quintic nonlinear terms, and their evolution with the energy of Hamiltonian system corresponding to the amplitudes. Based on the theory of plane dynamical system, we first make a detailed qualitative analysis to the plane dynamical system corresponding to the amplitudes of traveling wave solutions of the studied equation, then by applying the analysis method based on the first integral and several appropriate transformations, all seven families of elliptic function periodic wave solutions of the Zakharov equation are obtained. In addition, by studying the evolution limit of periodic wave solutions with respect to Hamilton energy and using the analysis method based on the first integral, all ten pairs of solitary wave solutions of the studied equations are also given under various parameter conditions. From the evolution analysis to the periodic wave solutions with respect to Hamilton energy, it can be seen that it is the energy H of the Hamiltonian system corresponding to the studied equation taking values in different ranges that makes the traveling wave solution of this equation appear as periodic wave solution or solitary wave solution.