General Traveling Wave Solutions of Nonlinear Conformable Fractional Sharma-Tasso-Olever Equations and Discussing the Effects of the Fractional Derivatives

Engineering Research Center of Heavy Machinery Ministry of Education, Taiyuan University of Science and Technology, Taiyuan 030024, China School of Mechanical Engineering, Taiyuan University of Science and Technology, Taiyuan 030024, China Shanxi Provincial Key Laboratory of Metallurgical Device Design Theory and Technology, Taiyuan University of Science and Technology, Taiyuan 030024, China School of Applied Science, Taiyuan University of Science and Technology, Taiyuan 030024, China School of Mechanical and Equipment Engineering, Hebei University of Engineering, Handan 056107, China

Recently, we have looked at some papers that obtain the exact solutions of the traveling wave of fractional-order equations and have illustrated some solitary wave solutions contained in the exact traveling wave solutions. These articles used different methods to obtain a large number of accurate traveling wave solutions in various forms. In order to better understand our work, we present the definitions and properties of the conformable fractional derivative as follows.
Given a function f : ½0,∞ → R. Then, the conformable fractional derivative of f of order 0 < α < 1 is defined as [8] T α f ð Þ t ð Þ = lim The conformable derivative is a fractal derivative [9]. According to the fractal derivative theory, the influence of environmental abnormal fluctuation on physical behavior is equivalent to the influence of fractal space-time transformation [10,11]. Several properties of conformable fractional derivative definition are as follows: These properties have been proved in literature [8], and there is no need to repeat them here. The physical interpretation of the conformable fractional derivative can be found in the literature [12].
According to Theorem 2.11 (chain Rule) in literature [13], we show the following properties of consistent fractional derivatives.
For more detailed knowledge, refer to literature [13]. Formula (6) is briefly proved as follows.
Proof. Set u = t + εt 1−α in the definition, and you get Now, let us look at some of the work with the STO equation of fractional order to find the exact traveling wave solution.
In literature [1], the author used fractional complex transformation for equation (1) and then used the expfunction method to obtain its precise traveling wave solution.
In literature [2], the author used fractional complex transformation on equation (1) and then used ðG ′ /G 2 Þ -expansion method to obtain the precise traveling wave solution of equation (1).
In literature [3], the author used fractional complex transformation on equation (1) and then used the novel ðG′/GÞ-expansion method to obtain the exact traveling wave solution of equation (1); the generalized Kudryashov method was also used to obtain the precise traveling wave solution of equation (1).
In literature [4], the author used the improved ðG′/GÞ -expansion method to get the exact traveling wave solution of equation (1) after using the complex fractional transformation.
In literature [5], the author used fractional complex transformation for equation (2), and then extended tanhcoth method was used to obtain the precise traveling wave solution of equation (2).
In literature [6], the author used fractional complex transformation for equation (3) and then used the Riccadi function expansion method to obtain the exact traveling wave solution of fractional-order equation (3).
In literature [7], the author used fractional complex transformation in equation (3) and then used a new generalized ðG′/GÞ-expansion method to obtain the precise traveling wave solution of the fractal equation (3).
Here, we have just listed some of the articles on the accurate line-wave solution to the STO equation of fractional order. It can be seen from literature [1][2][3][4][5][6][7] that the first step of these authors was to reduce the STO equation of the fractional order to a nonlinear ordinary differential equation by using fractional complex transformation and then to solve the reduced equation by various methods to obtain the solution of the original fractional-order equation. In addition to the methods provided in literature [1][2][3][4][5][6][7], there are other techniques that can be used to obtain wave solutions of different structures [14][15][16][17][18][19]. It can be seen from equations (1)-(3) that the most general equation is equation (1), so we only discuss the accurate general traveling wave solution of equation (1).
For equation (1), α, β are conformable fractional derivatives and the fractional complex transformation used is [20] where c, k are parameters to be determined. Through equation (8), equation (1) can be transformed into In literature [1][2][3][4][5][6][7], some articles integrate equation (13) once and then assume that the integral constant is zero, 2 Advances in Mathematical Physics which will lead to the exact solution obtained being less general than its original one [21,22]. Equation (9) is a thirdorder differential equation, and in general, its general solution should contain three arbitrary constants. From this perspective, the exact line-wave solutions obtained in the article [1][2][3][4][5][6][7] we mentioned are not general solutions, but only partial solutions. In this paper, we consider the integrability of equation (9) and combine with the fractional complex transformation to obtain the general traveling wave solution of the fractional STO equation, which is different from the exact solution obtained by using the function expansion method in known literature. As we all know, from the exact solutions of traveling waves, various forms of solutions can be obtained by choosing appropriate parameters, such as soliton solutions and periodic wave solutions. This conclusion can be obtained by bifurcation analysis of the first integral equation of equation (9) [23]. If the reader wants to know how the exact linear wave solutions of equation (9) are bifurcated and how the exact traveling wave solutions correspond to different forms of solutions, please refer to [23]. We selected from the general solution of traveling wave a kink solitary wave to fractional-order parameter change on the influence of the waveform, found that when two fractional-order parameters decrease at the same time, the kink soliton will become more smooth, but the location remains the same basic phenomenon, through the analysis found that the cause of this phenomenon is a kink soliton in the clockwise and counterclockwise to balance. Papers with such detailed analysis are rare.

The General Solution of the Fractional STO Equation in Space and Time in the Traveling Wave
Substituting equation (8) into equation (1) for transformation, we get Integrate equation (10) once, and we get Considering the specific function transformation, where } F ′ } = dF/dξ,F = FðξÞ, we can easily get the following equation.

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(1) When discriminant Δ = ðq/2Þ 2 + ðp/3Þ 3 = 0, that is, ðC 0 Þ 2 = 4c 3 /27ρk, the three roots of equation (17) are reduced to equation (16) has the following general solution: where C 1 , C 2 , and C 3 are arbitrary constants Without loss of generality, we can assume C 1 is not equal to zero. By substituting equation (22) into equation (12) and combining with complex transformation equation (8), the general traveling wave solution of the original equation (1) is expressed as In equation (23), there are only two arbitrary constants, which is caused by the fixed arbitrary constant C 0 . Without loss of generality, we could writeC 4 = C 2 /C 1 , C 5 = C 3 /C 1 .

The Discussion and Explanation
We observed some articles about the STO equation of fractional order and found that they more or less ignored the properties of the equation itself in the solution process, and the exact solution of the traveling wave obtained by various function expansion methods was not the general solution. Because in general, it makes sense that the precise general solution to equation (10) should contain three arbitrary constants. In the following part, we will find the kinked solitary wave solution from the general solution obtained in this paper and analyze the influence of the synchronous change of two fractional derivatives on the kinked solitary wave solution waveform and the reasons for this phenomenon.
Let ξ in equation (25) be the expression in equation (8), namely, We take the integral constant C 0 in equation (11) as zero; then, r 1 , r 2 , and r 3 in equation (25) are valued as

then, equation (25) is rewritten as
In equation (27)  As can be seen from Figure 1, as the fractional derivative α and β become smaller synchronically, the twist of the kink solitary wave becomes smoothed, but the position of the twist of the kink solitary wave basically remains at the straight line x = t on the t-x plane. In order to further explain this phenomenon and promote our previous work [24], this paper conducts a more detailed study of the influence of fractional derivative on the shape of kink solitary wave in two steps. First, the value of the time fractional derivative α is fixed, and only the change of the spatial fractional derivative β is observed. The result is shown in Figure 3. It can be seen from Figure 3 that the spatial fractional derivative β values 1, 0.7, and 0.4 in turn, and the kinked solitary wave rotates at the kinked position in the t-x plane, and the direction of rotation is close to the line t = 0 inside the t-x plane. Secondly, the value of the spatial fractional derivative β is fixed, and only the change of the time fractional derivative α is observed. The result is shown in Figure 4. It can be seen from Figure 4 that the time fractional derivative α is 1, 0.7, and 0.4 in turn, and the kinking position of the kink solitary wave in the t-x plane also rotates, with the direction approaching the line x = 0 in the x-t plane.
By comparing Figures 3 and 4, it is found that the spatial fractional derivative β becomes smaller, making the kink of the kink isolated wave as shown in the figure rotate clockwise in the t-x plane, while the time fractional derivative α becomes smaller, making the kink of the kink isolated wave as shown in the figure rotate counterclockwise in the t-x plane. When the fractional derivatives change synchronically, the kinks of the isolated wave in Figure 1 reach a balance in the clockwise rotation and counterclockwise rotation, and the position of the kinks almost stays the same. Meanwhile, the kinks of the isolated wave stretch together in the clockwise and counterclockwise rotation, which makes the kinks of the isolated wave smoothen.

Conclusion
The exact traveling wave solutions obtained by the function expansion method are usually not the general traveling wave solutions of fractional-order nonlinear partial differential equations. In this paper, the fractional-order STO equation is transformed into an integer-order STO equation through the complex fractional-order transformation, and then, the general traveling wave solution of the fractional-order STO equation is obtained through the transformation of a specific function, which will make our understanding of the traveling wave solution of the fractional-order STO equation more comprehensive. By setting parameters, a kinked solitary wave solution is extracted from the general traveling wave solution,

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and the influence of the fractional derivative on the kinked solitary wave is analyzed in detail. It is found that the kinked solitary wave becomes more smooth when the fractionalorder parameters are synchronized, but the position of the kinked solitary wave is basically unchanged. The position of the kinked solitary wave is basically unchanged because we have two fractional-order parameters, one of which becomes smaller so that the kinked waveform rotates clockwise, and the other fractional-order parameter becomes smaller so that the kinked waveform rotates counterclockwise. Such clockwise rotation and counterclockwise rotation achieve a balance. The kink solitary wave becomes smoother because of the tugging phenomenon that accompanies the process of reaching equilibrium.