Phase Portraits and Traveling Wave Solutions of a Fractional Generalized Reaction Duffing Equation

In this paper, we study the traveling wave solutions of the fractional generalized reaction Duffing equation, which contains several nonlinear conformable time fractional wave equations. By the dynamic system method, the phase portraits of the fractional generalized reaction Duffing equation are given, and all possible exact traveling wave solutions of the equation are obtained.


Introduction
. Furthermore, the exact soliton solutions of gRDM have been obtained by using the generalized hyperbolic function method, the Bäcklund transformation obtained by the homogeneous balance method, the first integration method of the fractional derivative in the sense of the improved Riemann-Liouville derivative, and the compatible fractional complex transformation method, respectively in [2] [3] [4] [5]. Based on an extended first-type elliptic sub-equation method and its algorithm, the new bell-shaped and kink-shaped solitary wave solutions, triangular periodic wave solutions and singular solutions of gRDM were solved in [6]. The accurate soliton solutions were obtained by using Bäcklund transformation of fractional Riccati equation, function variable method, and general projective Riccati equation [7] [8] [9]. In addition, other authors have used auxiliary function methods, Hermite transformation and Riccati equations, fractional sub-equations and other methods to study the exact solutions of gRDM in [10] [11] [12]. Recently, some new traveling wave solutions of the (2+1)-dimensional time-fractional Zoomeron equation and the superfield gardner equation have been obtained in [13] [14]. The fractional derivatives and fractional derivative equations have been deeply studied in [15] [16] [17] [18]. In this paper, we consider the following fractional order generalized reaction Duffing equation 2) Fractional Landau-Ginzburg-Higgs equation In this paper, we use the dynamic system approach [19] [20] to study the phase portraits and traveling wave solutions of the Equation (1), and try to construct all possible exact traveling wave solutions of this equation.

4) Fractional Duffing equation
The rest of this paper is organized as follows. In Section 2, we introduce some basic definitions and important properties of the fractional derivative. In Section 3, by applying the dynamic system approach [19] [20], we give the phase portraits of the Equation (1). In Section 4, we give all possible exact traveling wave solutions of the Equation (1) under different parameters. In Section 5, we state the main conclusions of this paper. mon fractional derivatives proposed by Khalil et al. [21]. Let Then, the conformal fractional derivative of f of order α is defined as . And the conformal fractional derivative has the following properties. Let , and , In addition, if f is differentiable, then

Phase Portraits of Equation (1)
Inspired by [22], we introduce the following fractional transformation where , k n are all arbitrary constants. According to (3)-(4), it infers By (6), substituting Equation (5) into Equation (1), we get ( ) where ' is the derivative with respect to ξ . Furthermore, it follows from [20] [23] that (7) is equivalent to the plane Hamiltonian system with the Hamiltonian ( ) In order to study the phase pictures of the system (8), it is necessary to study the equilibrium points of the system (8). Let the system has three equilibrium points ( ) . When 0 ∆ < , the system has only one equilibrium point , e e M U V be the coefficient matrix of the linearized system of the system (8) at an equilibrium point We have Trace By the planar dynamical theory [20], the above analysis and Maple, we obtain the following results and the phase portraits.
The corresponding phase portraits of the system (8) are shown in Figure 1.
is a center point.
is a center point.
are center points.
are saddle points.
is a saddle point.
are center points and 3 ,0 2 is a saddle point.
is a center point.
The corresponding phase portraits of the system (8)

Consider Case 1 in Section 3
By 0 ∆ = , it obtains Using the first equation of system (8) and equation (10), we get the following parameter expression Advances in Pure Mathematics 2) If 0 P < , 0 A < , corresponding to the homoclinic orbit (1) has a solution of shown in Figure 1(b). By By (8) and Equation (12), we get ( ) . The solution (13) is shown in Figure 1

1) If
Combining the first equation of system (8) and Equation (14), we have ( )   (17) is shown in Figure 6 , corresponding to the homoclinic orbit (1) has the solution of shown in Figure 2 or Figure 5. By It follows from (8) and (18) Using the first equation of system (8) and Equation (19) has the solution of shown in Figure 2 or The relation 4 5 6 7 U U U U U < < < < holds on the U-axis. Therefore, by using the first equation of system (8) and equation (20)

Consider Case 3 in Section 3
1) If 0 P < , 0 A < , corresponding to the homoclinic orbit (1) has the solution of shown in Figure 7(b). By Using the first equation of system (8) and Equation (21), we get the following parameter expression ( ) Using the first equation of system (8) and Equation (23), we get the following parameter expression ( )

Conclusion
In conclusion, we obtained the phase portraits of the traveling wave system by using the fractional complex transformation and the dynamical system method [19] [20]. Moreover, we construct all possible accurate traveling wave solutions of Equation (1) under different parameter conditions.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.