Research Article Analytical Solutions of the Fractional Complex Ginzburg-Landau Model Using Generalized Exponential Rational Function Method with Two Different Nonlinearities Ghazala Akram , 1 Maasoomah Sadaf , 1 M. Atta Ullah Khan, 1 and Hasan Hosseinzadeh 2 1 Department of Mathematics, University of the Punjab, Lahore 54590, Pakistan 2 Department of Mathematics, Ardabil Branch, Islamic Azad University, Ardabil, Iran Correspondence should be addressed to Hasan Hosseinzadeh; hasan_hz2003@yahoo.com Received 5 October 2022; Revised 13 February 2023; Accepted 22 February 2023; Published 13 March 2023 Academic Editor: Onur Alp Ilhan Copyright © 2023 Ghazala Akram et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The complex Ginzburg-Landau model appears in the mathematical description of wave propagation in nonlinear optics. In this paper, the fractional complex Ginzburg-Landau model is investigated using the generalized exponential rational function method. The Kerr law and parabolic law are considered to discuss the nonlinearity of the proposed model. The fractional effects are also included using a novel local fractional derivative of order α. Many novel solutions containing trigonometric functions, hyperbolic functions, and exponential functions are acquired using the generalized exponential rational function method. The 3D-surface graphs, 2D-contour graphs, density graphs, and 2D-line graphs of some retrieved solutions are plotted using Maple software. A variety of exact traveling wave solutions are reported including dark, bright, and kink soliton solutions. The nature of the optical solitons is demonstrated through the graphical representations of the acquired solutions for variation in the fractional order of derivative. It is hoped that the acquired solutions will aid in understanding the dynamics of the various physical phenomena and dynamical processes governed by the considered model. 1. Introduction Nonlinear partial differential equations (NPDEs) govern the majority of physical phenomena and dynamical processes. The study of nonlinear wave propagation problems provides motivations for developing NPDEs. The most significant task in nonlinear science is to obtain solutions for NPDEs, particularly solitary and soliton wave solutions. Nonlinear natural and physical phenomena are better understood with the aid of solutions. In chemistry, mathematical physics, mathematical biology, and many other basic sciences, NPDEs are frequently used to model different phenomena and dynamics. The NPDEs such as the Lakshmanan- Porsezian-Daniel model [1, 2], Navier-Stokes equations [3], shallow water-like equation [4], resonant nonlinear Schrö- dinger equation [5], and Korteweg-de Vries equation [6] are used to transform numerous natural processes in math- ematical form. Numerical simulations, mathematical expres- sions, and formulations are increasingly used in various problems of science in recent years [7–10]. In this paper, the complex Ginzburg-Landau (CGL) model can be expressed as iU t + δ 1 U xx + δ 2 F U j j 2 ÀÁ U − δ 3 U j j 2 U ∗ 2 U j j 2 U j j 2 ÀÁ xx − U j j 2 ÀÁ x À Á 2   − δ 4 U = 0, ð1Þ where U represents the profile of wave propagation and it depends upon the space variable x and time variable t . Hindawi Advances in Mathematical Physics Volume 2023, Article ID 9720612, 22 pages https://doi.org/10.1155/2023/9720612