Exact solitary and periodic wave solutions of high-order nonlinear Schrödinger equation and their relationship with Hamilton energy

Abstract

In this paper, we study the exact solitary wave solutions, periodic wave solutions, and bounded rational function solution of the high-order nonlinear Schrödinger equation and the evolutional relationships between the solitary and periodic wave solutions dependent on the Hamilton energy of their amplitude. First, based on the theory and the method of planar dynamical systems, we give a detailed qualitative analysis of the planar dynamical systems corresponding to the amplitude of traveling wave solutions. Then, based on the first integral of the system, we obtain the exact solitary wave solutions, periodic wave solutions, and bounded rational function solution of the equation in various forms by the analysis method, the integral technique, and proper transformation and establish the relationship between the solutions and the Hamilton energy of their amplitude. Furthermore, we discuss the evolutional relationships between the solitary and periodic wave solutions and reveal that the solitary and periodic wave solutions of the equation are essentially determined by the energy change in the Hamilton system corresponding to their amplitude. Finally, we give some diagrams that demonstrate the evolution from periodic wave solutions to solitary wave solutions when Hamilton energy changes.