New soliton solutions of the nonlinear Radhakrishnan-Kundu-Lakshmanan equation with the beta-derivative

In this paper, the modified exponential function method is applied to find the exact solutions of the Radhakrishnan-Kundu-Lakshmanan equation with Atangana’s conformable beta-derivative. For this, the definition of the conformable beta derivative proposed by Atangana and the properties of this derivative are firstly given. Then, the exact solutions of the nonlinear Radhakrishnan-Kundu-Lakshmanan equation which can be stated with the conformable beta-derivative of Atangana are obtained by using of the presented method. For the related problem in this paper, two solution cases are obtained, in each case five different solution families. The exact solutions found as a result of the application of the method seem to be 1-soliton solutions, dark soliton solutions, periodic soliton solutions and rational function solutions. According to the obtained results, it can be said that the Radhakrishnan-Kundu-Lakshmanan equation with Atangana’s conformable beta-derivative has different kinds of soliton solutions. Also, three-dimensional contour and density graphs and two-dimensional graphs drawn with different parameters are given of these new exact solutions. These graphs give detailed informations about the physical behavior of the real and imaginary parts of the exact solutions obtained.


Introduction
In recent years, many research papers have been made to find exact solutions of the problems that can be modeled mathematically by using fractional derivatives to understand some physical phenomena (Zhang and Zhang 2011;Guo et al. 2012;Pandir et al. 2013a;Demiray et al. 2014aDemiray et al. , 2014bGomez-Aguilar 2019, 2021a). Such physical phenomena are often explained by nonlinear FPDEs. Fractional differential equations have applications in many fields such as physics, dynamics, signal processing, control theory, continuum mechanics, solid-state physics, engineering, chemistry, biology. There are different definitions of fractional derivative operators in the literature. The most wellknown of these are Jumarie, Caputo, Riemann-Liouville, Liouville-Caputo, Caputo-Fabrizio, Atangana-Baleanu (Podlubny 1999;Fabrizio 2015, 2014;Jumarie 2006).
In 2014, a new fractional derivative was defined and called the conformable derivative (Khalil et al. 2014). All properties of this derivative have the fractional compound. Exact solutions of the many differential equations have been investigated using this new fractional derivative operator (Chen and Jiang 2018;Darvishi et al. 2018). Recently, many scientists have contributed to the development of this fractional derivative by doing research on the conformable derivative. Atangana et al. gave a new definition to this fractional derivative and named it beta-derivative (Atangana and Goufo 2014;Atangana and Oukouomi Noutchie 2014;Atangana 2015). The function in this new derivative depends on the range from which it is derivatived. Many equations involving this derivative are reported in some interesting studies Yepez-Martinez et al. 2018a, 2018bGhanbari and Gomez-Aguilar 2019;Demiray 2020;Hosseini et al. 2020aHosseini et al. , 2020bYepez-Martinez and Gomez-Aguilar 2021b;Tuluce Demiray and Bayrakci 2021;Akturk 2021). In these papers, time fractional Hunter-Saxton equation, Radhakrishnan-Kundu Lakshmanan equation with β-conformable time derivative, Biswas-Arshed equation with beta time derivative, the Sasa-Satsuma equation in the monomode optical fibers including the beta-derivatives, resonance nonlinear Schrödinger type equation with Atangana's-conformable derivative, space-time fractional Heisenberg ferromagnetic spin chain equation and many other similar non-linear problems are handled and these problems are respectively solved by using the first integral method, sub-equation method, the generalized exponential rational function method, generalized Kudryashov method and modified It is solved with exp(− Ω(η))-expansion function method.
In this paper, the validity of the modified exponential function method is researched to determine exact solutions of equations containing Atangana's conformable beta-derivative. This method has been applied to various nonlinear physical problems and has been found to be effective. The method proposed in this article can be used not only for fractional differential equations containing beta derivatives, but also to construct exact solutions of fractional order or integer order nonlinear partial differential equations defined with all other derivative operators. Because this method is based on the principle of reducing nonlinear partial differential equations to nonlinear ordinary differential equations by wave transform. Therefore, this method can be applied in all fractional or integer order partial differential equations that can be converted into ordinary differential equations.
The rest of the article is designed as follows: In Sect. 2, some basic properties of the conformable beta-derivative of Atangana are given. Then, the modified exponential function method is explained in detail for fractional partial differential equations with betaderivative. In Sect. 4, the application of the method is given. This article is completed with a conclusion.

Beta-derivatives
The fractional comformable beta-derivative in the mathematical model in this study is widely used in physical problems. The definition of this derivative is as follows. The most important reason for choosing the fractional derivative of Atangana for the mathematical model used in this study is that it can provide various properties of the fundamental derivatives. These properties are stated below: (i) Let a and b the real numbers. f and g ≠ 0 are differentiable functions with respect to beta in the interval (0, 1] . According to these conditions, the following feature is provided, (ii) Where c is any constant that satisfies the following equation, (1) and h → 0 , when → 0 , we obtain, together with is any constant and, accordingly, the following equation can be written

The modified exponential function method
In this section, obtaining exact solutions of mathematical models represented as Atangana derivatives using the modified exponential function method will be explained in detail (Rahman 2014).
According to this method, first of all, the general form of the studied nonlinear fractional differential equations should be determined. The u function and its derivative terms used in the mathematical model analyzed in the study are written as follows: where x space and t represents time. According to the method, after the general form of the investigated equation is obtained, the operations are performed according to the following steps: Step 1: In this part, the complex wave transform is arranged according to the independent variable that the solution function u in the nonlinear fractional differential equation depends on: where , and are constants. According to the method, if the derivative terms in Eq. (9), which is considered as the general form of the mathematical model, are obtained by using wave transform (10) and written instead, Step 2: In this part, the assumed function as the wave solution of the mathematical model according to the method is given as where E is the integration constant obtained by integrating Eq. (13). The terms of and are also constants that the method must provide to their family state.
Step 3: In this section, which is the last part, the limits of the symbols for the last part are determined in Eq. (13), which is accepted as the solution of the mathematical model. In order to determine this part, the balancing principle must be used. The operation of this technique is as follows: a relationship between n and m is obtained by balancing the term containing the highest order derivative in the nonlinear ordinary differential equation of the studied mathematical model with the term of the highest order. Then, the n value is obtained by giving an arbitrary value to the m term in the found relation. In this way, the limits of the solution function are determined. After all these operations, the u function required in Eq. (11) and the derivative terms of u with respect to are formed from Eq. (13) and written in their place. Then, by arranging the e ( ) term and its powers, a system of algebraic equations consisting of A 0 , A 1 , A 2 , ⋯ , A n , B 0 , B 1 , B 2 , ⋯ , B m is obtained. The obtained coefficients are replaced in the solution function and it is checked with the help of the package program that it (16) ( ) = − ln e ( +E) − 1 .
(18) ( ) = ln ( + E), provides the equation. Finally, by giving appropriate values to the parameters in the solution function determined to provide the equation, taking into account the family conditions specified in the method, the three-dimensional, density, contour graphics representing the behavior of the mathematical model and the two-dimensional graphics within the appropriate t time value are obtained using the package program.

Applications of the nonlinear Radhakrishnan-Kundu-Lakshmanan (RKL) equation with beta-derivatives
In According to the method used in the article, firstly, the following wave transform is applied by considering the independent variables in the RKL equation, which is investigated with beta derivatives. In this way, the nonlinear fractional differential equation is reduced to an ordinary differential equation. When the derivative concepts in Eq. (19) are obtained from the complex wave transformation equation and substituted into Eq. (19), the real and imaginary parts of this equation are respectively: The balance procedure is applied considering the obtained Eqs. (21) and (22). In other words, by equalizing the term u 3 with the highest order derivative and the nonlinear term u ′′ in these equations, the relation n = m + 1 is found regarding the m and n limits in Eq. (12). Then n = 2 is obtained by choosing m = 1 . In order to obtain the most basic solution functions and to test the method on the related problem, when m = 1 is taken, n = 2 is obtained according to the balance procedure.
Considering the m and n values obtained above, Eq. (12) is written as follows. Derivative terms required in Eq. (21) and (22) are obtained from Eq. (23) as follows; Then the following steps are followed so that Eqs. (19) and (20)  Considering all these steps, the simplest form of Eqs. (19) and (20) are as follows: If Eqs. (23-25) is replaced in Eq. (26) and the equation obtained is arranged according to the powers of e ( ) , the system of algebraic equations is found. By solving this system, the following cases are obtained.

Case 1
Family 1: When ≠ 0 and 2 − 4 > 0, Let's construct the graphs of this solution function by determining the appropriate parameters according to the family condition.

Results and discussion
We discuss the results to the nonlinear Radhakrishnan-Kundu-Lakshmanan (RKL) equation with Atangana's conformable beta-derivative determined by applying the modified exponential function method. When the exact solutions of RKL Eq. (19) are compared with the solutions obtained by Ghanbari and Aguilar (Ghanbari and Gomez-Aguilar 2019), we can say that there are different solutions. For this reason, the soliton solutions obtained by the modified exponential function method are new exact solutions that are not included in the literature. The obtained exact solutions u 1,1 (x, t) and u 2,1 (x, t) are called the dark soliton solution. The solutions of u 1,2 (x, t) and u 2,2 (x, t) are described as periodic soliton solutions. The solutions u 1,3 (x, t) and u 1,5 (x, t) are also entitled as 1-soliton solution and rational function solution, respectively. However, in paper (Ghanbari and Gomez-Aguilar 2019), the authors constructed some exponential and rational function solutions to the nonlinear Radhakrishnan-Kundu-Lakshmanan equation with Atangana's conformable beta-derivative. At the same time, three-dimensional contour and density graphs and two-dimensional graphs are plotted in Fig. 1, 2 , 3, 4, 5, 6, 7, 8, 9, 10 which represent with different parameters. These graphs are shown to help us understand the complex phenomena such as the propagation and dynamics of light pulses.

Conclusions
In this study modified exponential function method has been successfully applied to attain were plotted using Mathematica 12 according to the appropriate parameters. In the literature research, it was determined that the exact solutions obtained in the study were not found in the literature. In this paper, we have easily seen that the physical behavior of the wave types that can be explained by the solution functions we obtained and the behavior of the solution functions found in the articles published in the literature are different. This situation can also be seen in a simple way on 2 and 3 dimensional graphics. On the other hand, by substituting the obtained solution functions in the related differential equation, it is seen that the solutions satisfy the equation and are verified. With the analyzes on all these new and old studies, it can be shown that the results produced are new and that they have not been included in the literature before, and the appropriate problem and method can be determined for future studies.