New Exact Solutions of the Fractional Complex Ginzburg–Landau Equation

Chun Huang,Zhao Li,Maria L Gandarias

doi:10.1155/2021/6640086

Abstract

In this paper, we apply the complete discrimination system method to establish the exact solutions of the fractional complex Ginzburg–Landau equation in the sense of the conformable fractional derivative. Firstly, by the fractional traveling wave transformation, time-space fractional complex Ginzburg–Landau equation is reduced to an ordinary differential equation. Secondly, some new exact solutions are obtained by the complete discrimination system method of the three-order polynomial; these solutions include solitary wave solutions, rational function solutions, triangle function solutions, and Jacobian elliptic function solutions. Finally, two numerical simulations are imitated to explain the propagation of optical pulses in optic fibers. At the same time, the comparison between the previous results and our results are also given.

Highlights

It is well known that the Ginzburg–Landau equation is one of the most important models to describe optical phenomena [1,2,3,4,5,6,7]
A variety of powerful mathematical approaches have been developed to derive the exact solutions to Ginzburg–Landau equation, such as the (G′/G2)-expansion method [16], the Modified simple equation method [17], the F-expansion [18], the sineGordon expansion method [19], the extended direct algebraic method [20], and the dynamical system method [21,22,23]
Consider the following time-space fractional complex Ginzburg–Landau equation [24,25,26,27,28]: i zδu zt2δ

Summary

Introduction

It is well known that the Ginzburg–Landau equation is one of the most important models to describe optical phenomena [1,2,3,4,5,6,7]. Consider the following time-space fractional complex Ginzburg–Landau equation [24,25,26,27,28]:. In [24], Sulaiman et al studied the conformable time-space fractional complex Ginzburg–Landau equation via extended sine-Gordon equation expansion method. In [25], Abdou et al considered the fractional complex Ginzburg–Landau equation by employing the extended Jacobi elliptic function expansion method. The complete discrimination system method proposed by Liu is very powerful and useful tool, and the exact solutions of many fractional partial differential equations have been solved (see [29,30,31,32,33,34,35]). We employed the complete discrimination system method to construct new exact solutions of the fractional complex Ginzburg–Landau equation. Remark 1. e conformable fractional derivative possesses the following properties:

Complete Discrimination System for the Polynomial

Applications

Graphical Representation of the Obtained Solutions

Conclusion