Nonlinear Dynamics and Exact Traveling Wave Solutions of the Higher-Order Nonlinear Schrödinger Equation with Derivative Non-Kerr Nonlinear Terms

Abstract

By using the method of dynamical system, the exact travelling wave solutions of the higher-order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms are studied. Based on this method, all phase portraits of the system in the parametric space are given with the aid of the Maple software. All possible bounded travelling wave solutions, such as solitary wave solutions, kink and anti-kink wave solutions, and periodic travelling wave solutions, are obtained, respectively. The results presented in this paper improve the related previous conclusions.

The First Integral Method for Solving Exact Solutions of Two Higher Order Nonlinear Schrödinger Equations

“…Step 4: According to the qualitative theory of ODEs [30], if we can find the first integrals to equation (9) under the same conditions, then the general solutions to equation (9) can be found directly. However, there is no systematic theory that can tell us how to find its first integrals, nor we know what these first integrals are.…”

“…When we want to understand the principle of physical phenomena, the work to solve exact solutions of nonlinear PDEs and to research their properties is significative. In order to find the exact solutions of nonlinear PDEs, pioneers presented the following methods, such as tanh-sech function method [1], projective Riccati equation method [2], Kudryashov method [3], sine-cosine method [4], Jacobi elliptic function expansion method [5], F expansion method [6], exp-function method [7], Hirota bilinear method [8], bifurcation theory method of dynamic systems [9] and so on.…”