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.
Highlights
The nonlinear Schrödinger (NLS) equation is a kind of widely used nonlinear partial differential equation
In 1973, Hasegawa discussed the transmission of optical pulses in the fiber in Ref. 1, and when the group velocity dispersion of the fiber was balanced with the nonlinear effect, he deduced the NLS equation, iqT
When the parameter ratio β1 : β2 : β3 = 0:1:1, Eq (2) can be reduced to the first kind derivative nonlinear Schrödinger equation;7 when the parameter ratio β1 : β2 : β3 = 0:1:0, Eq (2) can be reduced to the second kind derivative nonlinear Schrödinger equation;8 when the parameter ratio β1 : β2 : β3 = 1:6:0, Eq (2) can scitation.org/journal/adv be reduced to the Hirota equation,9 which describes the propagation of short pulses in nonlinear optical fibers; when the parameter ratio β1 : β2 : β3 = 1:6:3, Eq (2) can be reduced to the Sasa–Satsuma equation,10 which describes the propagation of ultrashort pulses in optical fibers
Summary
The nonlinear Schrödinger (NLS) equation is a kind of widely used nonlinear partial differential equation. (2) The reasons why this paper uses the theory of the planar dynamic system and the analysis method based on the first integral of the system are mainly the following two considerations: (i) In the past, literature studies for solving the periodic wave solutions of elliptic functions rarely narrate the meaning of the modulus m, so it is difficult to understand why it is a variable and why we can get solitary wave solutions when m → 1. Scitation.org/journal/adv periodic wave solutions of the equation are essentially determined by the energy change in the Hamilton system corresponding to their amplitude It is meaningful for a nonlinear system; there could be various phenomena in a system, and only when we understand and master the basic factors can we apply and control the actual system corresponding to these equations. This is the second innovation of this paper compared with previous literature studies
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