New soliton solutions of kraenkel-manna-merle system with beta time derivative

This article discusses the fractional Kraenkel-Manna-Merle (KMM) system, which describes the motion of a nonlinear ultrashort wave pulse through saturated ferromagnetic materials with zero conductivity. The fractional behavior of this system was investigated using the beta derivative. The modified generalized exponential rational function method (MGERFM), developed by modifying the generalized exponential rational function method (GERFM), is applied to this system for the first time. Thus, some soliton solutions of the KMM system that have not been obtained before are presented for the first time in this study. In addition, 2D, 3D and density graphs of the obtained solutions for various values and ranges are presented. Discussions of these graphs are given and the found solutions are compared with other solutions.


Introduction
In recent research, it has been discovered by researchers that ferromagnetic materials have a wide range of applications in a wide variety of fields essential to everyday life, such as computers, generators, transformers, information and networking.Along with these studies, scientists have paid more attention to this field to meet the demands of massive data storage and massive data with large capacity and speed.For this purpose, various studies have been carried out to transform ferromagnetic materials into nanoscale materials with dimensions of 20-30 nm.With the help of large capacities and high densities but low volumes of data, processing, storage and distribution have been done [1][2][3].
In a study by Kraenkel et al in 2000, short-wave propagation in saturated ferromagnetic materials was investigated from Maxwell and Landau-Lifshitz-Gilbert equations, and they have obtained the system of equations given below [21], From the notations here, P shows dimensionless magnetic induction and K represent magnetization density.m and σ are constants and exemplify dimensionless saturation magnetization and Gilbert-Damping parameter respectively, ∇is divergence for vector field.Nguepjouo et al [22] introduced a combination of coordinate transformations and expansion series of the magnetization density in recent years, and thus changed the (1) system to the following form.where u = u(x, t) is magnetization and v = v(x, t) is external magnetic fields related to the ferrite.x and t denote displacement and time variables respectively.Coefficient s denotes the damping effect [1].
When the past studies for the fractional KMM system are examined, it is seen that the following studies have been carried out.Arshed et al obtain various solutions by performing the generalized projective riccati equations and modified auxiliary equation methods to the fractional KMM system [1].Zhang et al get Jacobi elliptic function solutions and solitary wave solutions by applying the Lie symmetry analysis method to the fractional KMM system [23].
In this present study, MGERFM was applied to the KMM system.Thus, various soliton solutions of this system have been obtained [24,25].

Definition of MGERFM
Step1: We consider nonlinear partial differential equation (NLPDE) given below: We first apply the wave transform given as below to equation (2); equation ( 2) is transformed into ordinary differential equation by using equation (3): where j and λ values that are not taken into account will be calculated later.
Step2: Assume that the solution of equation ( 5) is considered as: where p e p e .7 q q q q 1 2 3 4 Here value of M is determined through the homogeneous balance principle between the highest power nonlinear term and highest order derivative.p 1 , p 2 , p 3 , p 4 , q 1 , q 2 , q 3 , q 4 , a 0 , a i , b i , c i , (i = 1,K,M) constants are determined to fit the solution, such that the equation (6) verify the equation (5).
Step3: When equations (6) and (7) are placed in equation (5), a polynomial equation is obtained.A system of algebraic equations is obtained by setting each coefficient in the polynomial equal to zero.
Step4: If we solve the obtained system of equations and the found values consider in equation (5), the solutions of the discussed NLPDE are obtained.

Application of MGERFM
The following transformation is considered to find the solution of equation (1): It is obtained by applying the transform (8) to the system (2), By integrating equation equation (10), we get Substituting the equations (11) in (9), we get ( ) where θ is taken as the integration constant.By applying the homogenous balance principle to the highest power nonlinear term U 3 and highest order derivative U″ in equation (12), N = 1 is acquired.If N = 1 is taken into account in equation (6).
Considering the equalities in equation (18) in equations ( 13) and ( 14) the following solutions are obtained. , .
Inserting the these values in equations ( 13) and ( 14) the following solutions are found.
Considering the equalities in equations (28) in ( 13) and (24) the following solutions are found.  .
Embedding the these values in equations ( 13) and (24) the following solutions are obtained. .
Inserting the these values in equations ( 13) and (34) the following solutions are found.

( )
Placing the these values in equations ( 13) and (34) soliton solutions of system (2) are found. ( Considering the these values in equations ( 13) and (41) soliton solutions of system (2) are obtained.
Embedding the these values in equations ( 13) and (45) soliton solutions of system (2) are found.
Case 2:  Inserting the these values in equations ( 13) and (45) the following solutions are obtained.
Considering the these values in equations ( 13) and (55) soliton solutions of system (2) are obtained. .
Case 2: Inserting the these values in equations ( 13) and (55) the following solutions are found.

Results and discussion
In this study, semi-analytical solutions of the fractional KMM system with beta derivative are investigated.For this purpose, MGERFM, a new method created by modifying the generalized exponential rational function method, is applied.Thus, exponential function solution, complex exponential function solution, complex hyperbolic function solution and complex dark soliton solutions of the considered equation are obtained.In this The most important advantage of the method used in this study is that a wide variety of solution families can be created for each different value of p i and q i , (i = 1, 4) in equation (7).since it offers a wide range of solution families, it is a more general method than other methods.Thus, it is seen that this method is a useful method for finding the solution of any nonlinear partial differential equation.
Considering the previous studies on the fractional KMM system, the solutions u 10 (x, t) and u 12 (x, t) obtained in this study are similar to solutions (29) and (31) presented by Arshed et al [1].All other solutions have not been presented before according to our research and are presented for the first time in this study.To the best of our knowledge, no one has discussed exact solutions of the KMM system in the presence of damping effect with beta derivatives using MGERFM.
Figure 1 depicts the singular kink soliton of the real plot of u 1 (x, t) for the interval −4.5 < x < − 1.5 and −2.5 < t < − 1.5.Figure 2 describes the solitary wave of the imaginer plot of u 2 (x, t) for the interval 0 < x < 10 and 0 < t < 3. Figure 3 represent the singular kink soliton of the imaginer plot of u 3 (x, t) for the interval −5 < x < 5 and −2 < t < 2. Figure 4 delineates the singular soliton of the imaginer plot of u 4 (x, t) and the singular kink soliton of the real plot of u 4 (x, t) for the interval 0 < x < 2.6 and 0 < t < 2.6.Figure 5 shows the periodic traveling wave of the imaginer plot of u 5 (x, t) and the kink anti-kink solution of the real plot of u 5 (x, t) for the interval 0 < x < 5 and 0 < t < 5. Figure 6 portrays the solitary wave of imaginer and real plots of u 6 (x, t) for the interval −5 < x < 5 and −2 < t < 2. Figure 7 demonstrate the bell shape soliton of the imaginer plot of u 7 (x, t) for the interval 0.8 < x < 1.6 and 0.8 < t < 1.6.Figure 8 represent the singular periodic traveling wave of the imaginer plot of u 8 (x, t) for the interval −5 < x < 5 and −3 < t < 3. Figure 9 depicts the bell shape soliton of the imaginer plot of u 9 (x, t) for the interval 0.7 < x < 1.3 and 0.7 < t < 1.3.Figure 10 delineates the singular kink soliton of imaginer and real plots of u 10 (x, t) for the interval −2.5 < x < 2.5 and −2.5 < t < 2.5. Figure 11 describes the singular kink soliton of the real plot of u 11 (x, t) for the interval −1.5 < x < 1.5 and −1.5 < t < 1.5.Figure 12 shows the solitary wave of the real plot of u 12 (x, t) for the interval −1.5 < x < 1.5 and −1.5 < t < 1.5.Figure 13 depicts the singular periodic wave of the imaginer plot of u 13 (x, t) for the interval 5 < x < 10 and values.
5 < t < 10. Figure 14 represent the kink anti-kink solution of the real plot of u 14 (x, t) and the singular periodic wave of the imaginer plot of u 14 (x, t) for the interval 1 < x < 8 and 1 < t < 4. Figure 15 illustrates the kink antikink solution of the real plot of u 15 (x, t) and the singular wave of the imaginer plot of u 15 (x, t) for the interval 0 < x < 20 and 0 < t < 4. Figure 16 demonstrate the smooth soliton solution of the imaginer plot of u 16 (x, t) for the interval 3.5 < x < 11.5 and 3.5 < t < 5.

Conclusion
In this work, the fractional KMM system with beta time derivative is investigated.MGERFM, which is a modification of GERFM and recently introduced to the literature, is applied to this system for the first time.Thus, various soliton solutions of the studied system were found.Some of these solutions are presented for the first time in this study.In addition, various density, 3D and 2D graphical representations such as bell shape soliton, singular kink soliton, singular periodic wave etc were made by giving some values and intervals to these solutions.The method used in this study is applicable and effective for solving a wide range of nonlinear equations in mathematical physics.In addition, the most important advantage of this method is that it provides a wide variety of different solution families.values.