Chaotic behaviors, exotic solitons and exact solutions of a nonlinear Schrödinger-type equation

Abstract

In this paper, qualitative analysis is used to study a nonlinear Schrödinger-type equation. By applying traveling wave transform and choosing different parameter values, we obtain two types of dynamic systems. We apply the qualitative method to show that there are periodic solutions, optical solitary waves and exotic solitons for the equation. Then we perform the quantitative analysis to these different dynamic systems and obtain the corresponding exact solutions, which verifies the correctness of our previous conclusions. In addition, some new solutions are also constructed, such as, rational solutions and double period elliptic function solution. Finally, we add different perturbed forms to the dynamic system and obtain the largest Lyapunov exponents and corresponding phase diagrams, which demonstrates the existence of chaotic behaviors of the equation. To the best of our knowledge, the chaotic behaviors of the equation we obtained are firstly proposed.