Classification of All Single TravelingWave Solutions of Fractional Perturbed Gerdjikov–Ivanov Equation

+e fractional perturbed Gerdjikov–Ivanov (pGI) equation plays a momentous role in nonlinear fiber optics, especially in the application of photonic crystal fibers. Constructing traveling wave solutions to this equation is a very challenging task in physics and mathematics. In the current article, our main purpose is to give the classifications of traveling wave solutions of the fractional pGI equation. +ese results can help physicists to further explain the complex fractional pGI equation.

Among them, NLSE with cubic nonlinearity, namely, the perturbed Gerdjikov-Ivanov (pGI) equation, is the most studied [6][7][8][9][10][11], which is given by where u(t, x) is a complex valued wave profile with respect to time and space variables. u is the complex conjugation of u. c stands for the nonlinear dispersion coefficient. δ 1 , δ 2 , and θ are known parameters.
In recent years, with the development of fractional derivative, the fractional perturbed Gerdjikov-Ivanov equation has been widely concerned by many researchers in the following form: where u � u(t, x) represents unknown functions about time t and space x, respectively. u is the complex conjugation of u. D α t u and D α x u are the conformable fractional derivative [12][13][14].
In recent years, the traveling wave solution of the fractional pGI equation has attained a lot of interest in the filed of nonlinear science. Some explicit traveling wave solutions of the fractional pGI equation have been reported [15][16][17]. In Reference [15], Li and his collaborators studied the space-time fractional pGI equation by the fractional H-expansion method. In Reference [16], Ghanbari and Dumitru obtained the optical solutions of the fractional GI equation by using the generalized exponential rational function method. In Reference [17], Zulfiqar and Ahmad explored the optical solution of the fractional pGI equation by the Tanh method and Tanhcoth method, respectively. e main purpose of this paper is to give the classifications of traveling wave solutions of the fractional pGI equation by using the complete discrimination system method for polynomial, which is an important method to find the exact traveling wave solution of the fractional partial differential equation (FPDE). In general, the FPDE can be simplified to the following complete discriminant system by traveling wave transformation: where ϕ 1 , . . . , ϕ m are parameters. F(Ψ, ϕ 1 , . . . , ϕ m ) is thirdorder polynomial, fourth-order polynomial, or fifth-order polynomial. en, integrating equation (3) once, we obtain where ξ 0 is the integration constant. e readers can refer to [2,4,18] for details of this method. e structure of this paper is as follows. In Section 2, we simplify equation (2) to the nonlinear ordinary differential equation by fractional traveling wave transformation. en, the complete discrimination system is used to construct the classification of all single traveling wave solutions of the fractional pGI equation. In Section 3, we give a summary.

Conclusion
In the paper, the traveling wave solution of the fractional pGI equation is studied by using the complete discrimination system method. Firstly, we reduce equation (2) to the nonlinear ordinary differential equation by fractional traveling wave transformation. en, the complete discrimination system is used to construct the classification of all single traveling wave solutions of the fractional pGI equation. Among them, the implicit solutions have not been reported in other literatures. Compared with the existing literature [15][16][17], our solutions are more abundant, including rational function solutions, Jacobian function solutions, and implicit solutions. It is worth mentioning that the complete discrimination system method is an effective and powerful tool for constructing the traveling wave solution fractional pGI equation. Moreover, the method can be employed to construct the traveling wave solution of general FPDE.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.