The analysis of the soliton-type solutions of conformable equations by using generalized Kudryashov method

In this study, we applied the generalized Kudryashov method to two different conformable fractional differential equations and one system namely Burgers’ equation with conformable derivative, Wu-Zhang system with conformable derivative, and the conformable sinh-Gordon equation. Obtained solutions are soliton-type solutions. All obtained solutions have been verified and considered correct by placing them in the original equation with the help of the Maple software program. Also, we have plotted three-dimensional (3D) graphs of some of these solutions with the help of Maple.


Introduction
In the literature, it is well known that a broad range of problems in many research fields, such as wave propagation phenomenon, dynamical systems, mechanical engineering, fluid mechanics, biology, hydrodynamics, plasma physics, image processing, chemistry, optics, finance, and other fields of engineering and science. Nonlinear fractional partial differential equations (NFPDEs) have been proposed and searched by several scientists (Zubair et al. 2018;Raza et al. 2019;Hosseini et al. 2020). Many proposed different definitions have been introduced in the literature Park et al. 2020). Recently, researchers started to see incompleteness in most of the fractional derivative definitions (Samko et al. 1993;Kilbas et al. 2006). Since fractional differential equations have been a significant area in mathematical physics, nonlinear optics, mathematical biology, plasma physics,

Summary of conformable fractional derivative
Since the orientation of study in modeling real-world problems is shifting toward the use of fractional order derivative, there are various definitions of fractional derivatives in the literature for example Riemann-Liouville fractional derivative, Caputo fractional derivative, modified Riemann-Liouville fractional derivative, conformable derivative, beta derivative (Demiray 2020a, b) and so on. In this study, we have used the conformable fractional derivative, which is introduced by Khalil et al. The advantages of the conformable derivative are that the constant function is zero and the conformable fractional derivative behaves well in the product rule and chain rule.
The definition of conformable fractional derivative is given as follows: Assume that f ∶ [0, ∞) → R be a function. The conformable derivative of f of order , 0 < ≤ 1, is defined as for all t > 0. We can list some helpful features as follows: T (af + bg) = a(T f ) + b(T g) , for all a, b ∈ R T (fg) = fT (g) + gT (f ) Chain rule: Let f ∶ (0, ∞) → R be a differentiable and −differerentiable function, g be a differentiable function defined in the range of f.

The generalized Kudryashov method
We take into consideration a general CPDE of the formula (Kudryashov 2012;Bulut et al. 2014): The polynomial and derivates of u = u(x, t) are represented by F, where the nonlinear terms and the highest order derivatives are comprised. In a suitable manner of the following travelling wave transformation: where l represents the velocity of the wave, Eq. (1) is reduced thereby forming an ordinary differential equation (ODE) in the form We note that, in Eq. (3) the differentiation of u with respect to is represented by prime. We will integrate all the terms in Eq. (3). Conforming to this technique, the desired solution for the reduced equation is formed by a polynomial in R( ) as where a i (i = 0, 1, … , n), b j (j = 0, 1, … , m) are constants to be found ( a N ≠ 0 , b M ≠ 0 ) and Q = Q( ) is the solution of The solution of Eq. (5) written as Based on the homogeneous balance principle, one can find the positive integers N and M in Eq. (4) with the use of the Eq. (3). Namely, they can be determined by using homogeneous balance between the nonlinear terms and the highest order derivatives appearing in Eq.
(3 ). Finally, we can obtain a polynomial of R by subrogating Eqs. (4) into (3) along with Eq. (5). Here, we equate all the coefficients of polynomial R to zero to obtain an algebraic equation system. Solutions of this system using the assistance of the computer software gives the values of a i (i = 0, 1, … , n), b j (j = 0, 1, … , m) . Lastly, we find the soliton-type solutions of the reduced Eq. (3) by subrogating these acquired values and Eqs. (5) into (4).
There are different types of Kudryashov methods in the literature such as the Kudryashov method, modified Kudryashov method, generalized Kudryashov method, extended Kudryashov method. There are differences among these methods, such as auxiliary equations and the form of the desired solution. We can give a comparison of the methods as follows.
(i) for the classical Kudryashov method, the solution of the auxiliary equation is We have used the generalized Kudryashov method in this paper, since we seek the solution in a more comprehensive form.

Implementations
In the following section, three equations are implemented to show how the method works.

Burgers' equation with conformable derivative
Burgers' equation was firstly presented in 1918 by Bateman has been used as a mathematical model in several fields such as gas dynamics, number theory, elasticity theory, heat conduction, hydrodynamic waves, shock wave theory, elastic waves, termaviscous fluids, and turbulence theory (Bateman 1915). Many scientists have been worked on obtaining numerical and exact solutions for not only Burgers' equation (Burger 1939(Burger , 1948 where (x, t) the solution of the heat equation Eq. (7) and the derivative is an -order conformable fractional derivative. The numerical solution of fractional Burgers' equation has been acquired (Esen and Tasbozan 2016;Esen et al. 2013) implemented HAM to procure the approximate analytical solution of fractional Burgers' equation. Also, exact solutions to this equation have been founded by using the Ricatti expansion method (Abdel-Salam et al. 2014). Kurt et al. (2015) applied the homotopy analysis method to this equation and they utilized the Hopf-Cole transform (Kurt et al. 2016). Auto-Bäcklund transform and exact solutions to local conformable time-fractional viscous Burgers system have been found (Huang and Yang 2019). Some new travelling wave solutions for the one-dimensional Burgers equation were obtained (Cenesiz et al. 2017). Also, Demiray et al. applied the generalized Kudryashov method to time-fractional Burgers equation with Riemann-Liouville sense (Demiray et al. 2014a).
Using the following travelling wave transformation we can reduce Eq. (6) to the following ODE Integrating Eq. (9) with respect to once, and taking the integration constant as zero, we find the following equation According to the homogeneous balance principle, balancing the nonlinear term u 2 with the highest order nonlinear term, it can be founded as By setting M = 1 , we get N = 2. So the solution can be expressed as where R = R( ) is the solution of the Eq. (5). Based on this, we substitute Eqs. (11) into (10) and use Eq. (5). Afterwards, we equate all coefficients of the functions R k to zero. Therefore the following equation system can be obtained. Here a 0 , a 1 , a 2 , b 0 , and b 1 are parameters.

Case 3:
Then, by subrogating the acquisite values into Eq. (11) with Eq. (8), we get the soliton-type solution of the conformable Burgers' equation as follows .

Wu-Zhang system with conformable derivative
Wu-Zhang system demonstrates dispersive long waves in two horizontal directions on shallow waters which means that the pure plane has several speeds propagation that makes some of the waves spread outwards in space. A proper comprehension of all . Fig. 8 The bright soliton solution of the Wu-Zhang system with conformable derivative v 3,4 within the interval −10 ≤ x ≤ 10, 0 ≤ t ≤ 2, when C 1 = 1, = 0.9 Fig. 9 The kinky periodic solitary wave solution of the Wu-Zhang system with conformable derivative u 5,6 within the interval −10 ≤ x ≤ 10, 0 ≤ t ≤ 3, when C 1 = 4, = 0.9 solutions is useful for civil and coastal engineers in order to implement the nonlinear water wave model in the coastal and harbor design.

Fig. 12
The kinky periodic solitary wave solution of the Wu-Zhang system with conformable derivative v 7,8 within the interval −10 ≤ x ≤ 10, 0 ≤ t ≤ 2, when C 1 = 4, = 0.9 Here u and v represent the elevation and the surface velocity of water, respectively.
In the year 1996, Wu and Zhang (1996) proposed three sets of equations to model long, nonlinear, scattered gravitational waves that describe waves travelling in two horizontal directions over uniform shallow water depth. After doing some transformations and reductions, a dimensional long wave splitter (1 + 1) equation, known as the Wu-Zhang system has been acquired. These equations (Zheng et al. 2003) are advantageous for harbor and coastal designs in civil and coastal engineering.
While there is a gap in the Wu-Zhang system's harmonious time zone literature, the classical Wu-Zhang system is thought to have achieved soliton solutions by many researchers (Du et al. 2018;Mirzazadeh et al. 2017;Li et al. 2000). In recent years, this system of equations has been discussed by many researchers to find exact solutions of different types. Let us solve the system above based on the method implemented. We firstly take a travelling wave transformation as follows: The sytem Eqs. (16) is reduced the following ODE with use of Eq. (17): Integrating first equation of Eq. (18) with respect to once and setting the constant of the integration to zero, we get the following equation: The following equation can be founded by subrogating Eq. (19) into the second equation of Eq. (18): If we integrate Eq. (20) with respect to once, we find Then we balance the highest order derivative term u ′′ with the nonlinear term u 3 , we find (21) u �� = 3c 2 u − 9 2 cu 2 + 3 2 u 3 .
By setting M = 1, we find N = 2. Then, the desired exact solution becomes where R = R( ) is the solution of the Eq. (5). Based on this, we get the following equation system by subrogating Eqs. (22) into ( 21) along with Eq. (5) and then equating all coefficients of the functions R k to zero: and a 0 , a 1 , a 2 , b 0 , b 1 are parameters to be found. Solving these algebraic equations with the assistance of Maple, we attain the following different cases (Figs. 5,6,7,8,9,10,11,and 12): Case 1: By subrogating the acquisite values into Eq. (22), we attain the soliton-type solutions of the system as and Case 2: By subrogating the acquisite values into Eq. (22), we acquire the same soliton-type solutions of the Wu-Zhang system with conformable derivative in Case1. Case 5: By subrogating the acquisite values into Eq. (22), we acquire the solitary wave solutions of the Wu-Zhang system with conformable derivative as follows and For all the cases above C 1 is an integration constant.

The conformable sinh-Gordon equation
In this subsection, we are going to consider the conformable sinh-Gordon equation, which plays a crucial role in many scientific applications such as nonlinear optics, fluid dynamics, solid-state physics, nonlinear optics, kink dynamics, mathematical biology, plasma physics, kink dynamics, quantum field theory, and chemical kinetics (Tajadodi et al. 2021).
Using the following travelling wave transformation (29) Fig. 13 The multi-soliton interactions of the conformable sinh-Gordon equation derivative u 1 within the interval −10 ≤ x ≤ 10, 0 ≤ t ≤ 2, when C 1 = 1, = 0.8  (5). Afterwards, we equate all coefficients of the functions R k to zero. Therefore the following equation system can be obtained. Here a 0 , a 1 , a 2 , a 3 , b 0 , and b 1 are parameters.