Various wave solutions to the nonlinear fractional Korteweg-de Vries Zakharov-Kuznetsov equation by a new approach to the two-variable expansion scheme

Solving nonlinear differential equations is crucial for the design, optimization, and characterization of engineering systems. However, investigating such equations poses significant challenges and requires the development of new mathematical and computational techniques. In this article, we extend the two-variable (G′/G,1/G) -expansion approach and establish scores of general analytical solutions, consisting of different functions and arbitrary parameters, to the time-fractional Korteweg–de Vries Zakharov-Kuznetsov (KdV-ZK) equation. The KdV-ZK equation provides a powerful model for modulating nonlinear waves in an assortment of physical systems, including plasma physics, atmospheric science, fluid dynamics, soliton theory, and optical fibers. The proposed technique allows extracting several novel solutions to the KdV-ZK equation, which is essential for gaining new insights into physical problems. In addition, the suggested extension shortens the calculation process and enhances computational efficiency. The obtained solitons have potential applications in several scientific and technological fields. The beta fractional derivative and the consistent wave variable are considered to restructure the stated fractional nonlinear equation. This study investigates the merits and demerits of the fractional-order derivative β on the dynamics of the system through the representation of two-dimensional graphs, which vary according to different values of β. The enhanced computational efficiency might assist the researchers in exploring a wider range of phenomena, facilitating further extensive investigations, and contributing to overall progress in the fields of science and engineering.


Introduction
Most natural occurrences are expressed through nonlinear models since the universe is instinctively nonlinear.Nonlinear equations are effectively used to develop mathematical models of these natural problems.To overcome and investigate the phenomena arising in real life, analytical solutions play a vital role.In recent years, fractional nonlinear evolution equations (FNLEEs) and their solutions have significantly come into effect compared to integer-order nonlinear partial differential equations to examine insights into intricate phenomena ascending in scientific and technological fields like population dynamics, neuroscience, aerodynamics, plasmaphysics, optical fiber, etc That is why FNLEEs have become an inquisitive topic of research.Several academics made their best effort to look into the soliton solutions of numerous significant FNLEEs.Nonlinear models experience a solution, named soliton, through an assessment that equates the highest-ordered terms of linear and nonlinear components contained in the restructured equations.Because it has some significant properties, such as long-distance transmission capability with an unchanged shape, the soliton solution has become a burning topic of study for many researchers.As an illustration, notable literature on solitons includes works by Seadawy et al (2020), Arshad et al (2021Arshad et al ( , 2023)) Academics are currently focusing a lot on developing new and more effective techniques and have been able to establish several techniques, such as the tanh-function method (Evans andRaslan 2004, Wazwaz 2004a), the Backlund transformation (Omote and Wadati 1981, Wang and Wang 2001, Liu et al 2019, Han and Bao 2021), the sine-cosine method (Wazwaz 2004b), the modified rational sine-cosine approach (Ali et al 2022), the bilinear transformation scheme (Tanoğlu 2007, Xu andChow 2016) ), etc Most of the analytical approaches are generated followed the same process.The main points of divergences in these methods involve constructing individual trial solutions and using a variety of auxiliary equations, the general solutions of which are employed to originate novel solutions.To expand the previously established approach, academics modified either the trial solution or considered different auxiliary equations.For example, the ¢ G G ( ) / -exppansion method (Bekir 2008) considered trial solution in the form ) the auxiliary equation method (Pinar 2020) considered trial solution in the form The ( ¢ G G G , 1 / / )-expansion scheme was projected in 2010 and put into practice to unravel the Zakharov equation by Li et al (Li et al 2010).Many researchers adopted this method to delve the analytical soliton solution of significant FNLDEs, such as (Zayed and Alurrfi 2014, Miah et al 2017, Yokuş et al 2020, Duran 2021), and many others applied this method for solving significant nonlinear evolution equations.In former research, this method provided effective and reliable analytical exact solutions.This method is a result of modification of the ¢ G G ( ) / -expansion method.In 2008, Wang et al (Wang et al 2008) introduced a new approach named the ¢ G G ( ) / -expansion method.Since the introduction of this technique, several academics have made their best effort to modify the ¢ G G ( ) / -expansion method and developed some distinct approaches.The trial solution of the ¢ G G G , 1 ( ) / / -expansion scheme is expressed as where z G ( ) is the general solution to the equation / -expansion, the degree of the term G 1 ( ) / must be constrained within 1 (one).In this study, we proposed a modification of the ¢ G G G , 1 ( ) / / -expansion approach, which generate a new trial solution of the form å å The trial solution in the proposed scheme is distinctive, thus, this approach will generate some substantial and unparalleled solutions.
In this article, we suggest a new approach of the two-variable ¢ G G G , 1 ( / / )-expansion method and implement this approach to the time-fractional (3+1)-dimensional Korteweg-de Vries Zakharov-Kuznetsov (KdV-ZK) equation (Guner et al 2017) in the sense of beta derivative: , where a and c are self-steeping and dispersive factors, respectively.The time-fractional (3+1)-dimensional KdV-ZK model (1.1) was established to designate acoustic wave propagation through plasma of hot and cool electrons and fluid ions species.b D r t stands for the beta fractional derivative of the wave function = r r x y z t , , , ( ) with respect to temporal variable t.The KdV-ZK equation is a nonlinear equation that was introduced in the late 19th century.This equation is a modification of the KdV equation.The KdV equation was derived in 1895 by the Dutch mathematician Diederik Johannes Korteweg and his student Gustav de Vries (Miles 1981) to describe the propagation of shallow water waves in canals and rivers.In 1970s, the Russian physicist and mathematician Vladimir Zakharov and Vladislav Kuznetsov (Zakharov and Kuznetsov 1974) added a term to the KdV equation, which led to the creation of KdV-ZK equation.The (3+1)-dimensional KdV-ZK equation was introduced in 1990s in an effort to extend the KdV-ZK equation to higher dimensions.In recent years, the aforementioned equation has become the significant equation in the field of mathematical physics due to its wide range of applications and its ability to model nonlinear waves in four dimensions.The (3+1)-dimensional KdV-ZK equation provides a powerful tool for modeling nonlinear wave in a variety of physical systems, including plasma physics, fluid dynamics, soliton theory, optical fibers, and atmospheric science.This equation can exhibit the chaotic behavior, supports the formation of multi-soliton solutions, and can be solved both analytically and numerically.The chaotic behavior is important for modeling the behavior of waves in complex and turbulent system.Additionally, the mentioned equation satisfies several conservative laws, including conservation of energy, momentum, and mass.This equation has been investigated by several academics with the implementation of a number of approaches.As for example, Sahoo and Ray (Sahoo and Ray 2015), Kaplan and Bekir (Kaplan and Bekir 2016), Chen and Jiang (Chen and Jiang 2018), Shawba et al (2021), Zhao et al (2020), and many others left significant mark in this regard.
The motivation of this study is to establish a modified, trustworthy, and straightforward approach that can provide novel soliton solutions for NLEEs, which cannot be offered by any other existing methods.The suggested technique is a modification of the / -expansion approach, in which the trail solution has been expressed in a modified form to extract varied solutions.As the afore-mentioned KdV-ZK equation has projecting implication in study soliton theory, we have taken into account this equation to examine reliability of the proposed technique.Nowadays fractional calculus is being widely utilized to explore novel characteristics of solitons.Fractional models enable us to oversee physical phenomenon more precisely and with more degrees of freedom.It has a great impact on the formation of varied shape of solitons with high intensity.Therefore, we have considered the fractional form of the mentioned model in order to inspecting the reliability of the proposed technique for fractional differential models.
The subsequent sections of the article are structured as follows: In section 2, a clear definition of the beta derivative is provided, along with its notable properties.The essential steps of the method are delineated in section 3. Section 4 employs the proposed scheme to seek analytical soliton solutions for the given equation.The discussion and the comparison between the two variables, namely the / -expansion technique and the suggested approach, as well as the evaluation of the achieved results in relation to the previous findings, are presented in section 5.
Graphical representations of the obtained estimations have been discussed in section 6.Finally, we reach the conclusion presented in section 7.

Beta fractional derivative
The fundamental mathematical formula to define beta fractional derivative of a function U r ( ) can be expressed as (Atangana and Alqahtani 2016, Khatun and Akbar 2023): The regraded beta derivative has a number of postulates that are given below: 1.
The well-established beta derivative satisfies all the fundamental rules of the integer-order derivative, while all other definitions of fractional derivative have some shortcomings.

Key steps of the method
In this module, we will briefly describe the suggested technique.At the onset, contemplate a second order equation of following structure: / and l, m are constants.
Furthermore, from the above postulation, it can be derived as Depending on the diversity of conditions for the parameter l, equation (3.1) offers three categories of solutions encompassing hyperbolic, trigonometric, and rational functions.Hyperbolic form: For l < 0, the ensuing solution of equation (3.1) contains hyperbolic functions as follows.
where g , 1 g 2 are arbitrary parameters, and s = g g .Trigonometric form: For l > 0, the resulting solution of equation (3.1) is comprised of trigonometric functions as follows: where g , 1 g 2 are arbitrary parameters, and Rational form: For l = 0, the resultant solution of equation (3.1) consists of rational functions as follows: where g 1 and g 2 are arbitrary parameters.Now, suppose the equation to be solved has the following form: 3.9 x t x x tx tt t x t t xx x 1 K and r , t r , x 1 K are the beta fractional differentiations and the partial differentiations of r.
The pivotal procedures to employ the anticipated scheme are deliberated as follows: 1 st Phase (reduction to ordinary equation): The concerned equation has to be changed to an ordinary differential equation by attributing a variable transformation.For beta-fractional equation, the transformation is z ¼ = r t x x q , , , , Transformation equation (3.10) is needful for real equations, and equation (3.11) is needful for complex equations.z q( ) defines the amplitude function and q ( )describes the phase function.J and v stand for the wave number and speed of the wave function.
3.12 ( ) ( ) Here, P refers to a polynomial of the wave variable = q q x t , ( ) and its derivatives.2 nd Phase: We propose a trial solution of equation (3.12) holding the successive mathematical structure: w h e r e , 1 .

. 1 3
In solution (3.13), m , h n , h and r are unknown parameters.The value of r is determined with the assistance of the homogeneous balancing concept.For the negative or fractional values of r, we will put into use the subsequent transformation: Setting (3.14) into equation (3.12) yields a polynomial that assigns an integer value of r.
3 rd Phase: Putting z q( ) and its derivatives in equation (3.12), a polynomial containing f , ) including the parameters m , h n h will be initiated.Introducing the value of f¢, y¢, f 2 in the attained equation and then computing the components of ) to zero, the required algebraic system of equations can be constructed.
4 th Phase: At last, the arbitrary parameters containing in trial solution have to be obtained by resolving the constructed systems.Making use of the resultant outcomes, solutions of the considered model can be generated from equation (3.13).

Solutions to the KdV-ZK equation
Introducing the beta wave variable transform (3.10) to equation (1.1) yields the following nonlinear equation: where ( ) v is the speed of the wave.
Integration of equation (4.1) with respect to z leads to where k is an integral constant.
Recognizing the hypothesis of balancing principle, the accomplished result is = r 2. Therefore, the contemplated solution of equation (4.2) can be structured as: In equation (4.3), m , 0 m , 1 m , 2 n , 1 n 2 are unknown constants to be determined, and Performing the elementary procedure of the offered scheme, we attain several arrays of solutions for different values of l.
When l < 0, we attain solutions containing hyperbolic functions as follows: 1 2 0 0 ( ) Setting the parametric values assembled in Result 1 into solution (4.3) and using (3.10), we competently obtain the following solution: ) and g , 1 g 2 are constants.If we set the integral constant = k 0, we attain = m 0, along with Moreover, if m = 0, solution (4.5) becomes of the form as given below, If l > 0, the gained solutions will contain trigonometric functions as follows: along with z = + + - where Furthermore, for m = 0, solution (4.8) becomes For l = 0, we attain the solutions which contain rational functions.
Result 3: Using result 4, the attained solution can be presented as

By means of Result 3, the attained solution becomes
( ) and g , 1 g 2 are constants.

Comparison
In this section, we interpret the variance between the ¢ G G, ( / G 1 ) / -expansion process and the proposed technique.The comparison between the results of the considered equation established by putting into use the suggested scheme and the results reported in the literature will be associated in tables 1 and 2 in order to assess the novelty, compatibility, and reliability of the suggested technique.
For = r 1 (balanced value), both of these approaches deliver the same outcomes.Moreover, the soliton solutions of the governing equation attained in this article are different from the solutions offered by Al-Shawba et al (2018), which were attained by implementing the ¢ G G G , 1 ( ) / / -expansion process.In addition, the mathematical forms of the gained soliton solutions of the time-fractional (3+1)-dimensional KdV-ZK equation in this letter do not align with any solutions found in previous literatures that were solved using different approaches.However, some of the derived solutions exhibit the same physical characteristics as the previous

G) / -expansion technique
According to the methodology of this approach, trial solutions have the form.
According to the methodology of this approach, trial solutions have the form.
å å ( ) To execute this method, we must confine the order of y between 0 and 1 by using the value of y . 2 To execute the suggested technique, we must confine the order of f between 0 and 1 by using the value of f .

2
This method takes much time for execution as we have to implement the value of y 2 repeatedly in the equation to extinct the higher order of y.
This scheme takes less time for execution as we have to implement the value of f 2 once in the equation to extinct the higher order of f. and m = 0, solution (4.4) becomes For m = 0 and ) } where A, B, and m 0 are free parameters.
where C, D, and l 0 are free parameters.If we set m = 0 and = g 0, 2 the solution (4.4) can be articulated as For m = 0 and = A 0, 2 the solution (4.28) of this study can be expressed in simplified form as ) } where A, B, and m 0 are free parameters.
where C, D, and l 0 are free parameters.Setting m = 0 and = g 0, 1 the solution (4.7) can be written in the ensuing form.
For m = 0 and )} where A, B, and m 0 are free parameters.
where C, D, and l 0 are free parameters.For m = 0 and = g 0, 2 the solution (4.7) can be expressed as For m = 0 and = A 0, 2 the solution (4.32) of this study can be expressed in simplified form as )} where A, B, and m 0 are free parameters.
where C, D, and l 0 are free parameters.
solutions, for example, singular bell-shaped soliton, periodic soliton, , bell-shaped soliton, and singular periodic soliton.Although some of solutions are similar in behavior, their velocity, amplitude, and graphical representations differ.In addition, this paper presents several new types of solitons.Thus, it might be asserted that the obtained soliton solutions are novel.
The complexity of the method arises when the derived nonlinear equation becomes higher order.The / -expansion method provides potential and reliable solutions for the second-ordered equations.However, for several cases it might provide trustworthy solutions for the higher-ordered equations also.In case of higher order evolution equations, several methods considered the integrating constant to be zero to make the computational process facile.But, the last condition of the proposed approach is applicable for only nonzero integrating constant, even though the integrating constant can be taken as zero for the first two conditions.Considering integrating constant to be zero limited the abundancy of the obtained solutions, and sometimes the third condition fails to yield desired solutions.

Graphical descriptions
This module provides the graphical descriptions of the developed soliton solutions for the fitting definite estimates of the arbitrary parameters constituting the solutions.The graphs of the solutions have been mapped considering the span of axes -  x 10 10 and   t 0 10 as shown following figures 1-2.It is simpler to elucidate the consequence of a fractional derivative using a 2D plot.
The solution   characteristics are essential for a soliton to propagate long distance with unaltered shape.Bell-shaped soliton has significant application in optical fibers to improve signal quality and other optical instruments.Like other solitons, periodic solitons hold long life-span with unchanged properties and shape.It can be identified by its individual nature of periodic pulse propagation.Periodic solitons can be observed in the signal and blood processing system of living species.Hence, it has remarkable utilization in bioscience.Additionally, its stability and long-lasting properties make it suitable for the study of optical fiber communications.
The solution (4.9) presents singular periodic soliton for the considered values = -        chosen values of parameters.In spite of holding significance in understanding mathematical modeling of physical problems, singular solitons have narrow scope in physical applications.The self-steeping parameter can affect the shape of solitary wave profile due to nonlinear effects, depending on whether the parameter is positive or negative.The effect of self-steeping varies depending on positive or negative values of parameter, meanwhile, the dispersive terms become prominent for large absolute values of parameter.
From figure 10(a), it is observable that if the value of a increases, the trailing edge of the wave profile becomes steeper, and the intensity becomes higher at the peak of the pulse.Dispersion is a linear term that makes broaden of the wave profile, which lead to distortion of the pulse.Figure 10(b) shows that, dispersion coefficient serve as opposite to the self-steeping term.If the absolute value of dispersion parameter is large, dispersive effects become more prominent, leading to narrower and more spread-out solitary wave profiles.Hence, introducing balance between these terms allow us to obtain soliton solutions.

Conclusion
In the present study , we have successfully ascertained assorted useful analytical soliton solutions to the timefractional KdV-ZK equation employing the modified two-variable ¢ G G G , 1 ( ) / / -expansion approach.The equation under consideration is of significant importance for exploring soliton theory, optical fiber communications, fluid dynamics, plasma physics, and various other contexts.The solutions obtained from the equation yield a variety of highly conventional solitons, including periodic solitons, bell-shaped solitons, singular kinks, anti-bell-shaped solitons, kinks, and singular periodic solitons, for appropriate parameter values.The obtained soliton solutions remain unaltered in all dimensions of space as shown in the first figure.The comparison section supports the consistency of the proposed techniques to provide novel solutions.The solutions obtained through the introduced approach do not match to the solutions obtained via the original ( ¢ G G G , 1 / / )-expansion approach or any other techniques.Therefore, this approach might provide new path to investigate the nonlinear evolution equations and enhance understanding of the complex phenomena.Although the proposed scheme has a shortcoming in executing to solve higher order integrable NLEEs, it is worth noting that this technique can assist in finding potential solutions to FNLEEs equations whose exact solutions cannot be obtained using other methods.Therefore, to confirm the applicability, reliability, and effectiveness of the suggested technique, other FNLEEs need to be investigated, and this is our next focus.
, the modified exp-function method (Usman et al 2013, Shakeel et al 2023), the auxiliary equation technique (Sirendaoreji 2003, Akbulut and Kaplan 2018approach ( Khatun and Akbar, 2023, Al-Shawba et al 2021, Akbar et al 2023), the ¢ G G ( ) / -expansion method (Bekir 2008), the improved F-expansion technique(Akbar and Ali 2017), the generalized ¢ G G ( ) / -expansion method (Manafian et al 2017, Mohanty et al 2022, Mohanty et al 2023), the generalized and improved ¢ G G ( ) / -expansion method (Akbar et al 2012, Naher and Abdullah 2014), the improve ¢ G G ( ) / -expansion scheme (Zhang et al 2010, Liu and Zeng 2015, Sahoo et al 2020), the Bäcklund transformation (Yin et al 2022),the Riccati projective equation scheme (Zhao et al 2022), the similarity transformation method (Liu et al 2022), the ECIMM algorithm (Yin et al 2021), the optimal control strategies (Lü et al 2021), the fractional sub-equation method (Gepreel and Al-Thobaiti 2023, Alzaidy 2013 which is displayed in figure 1.It is acquainted as bright-dark mixed soliton, which propagates in a long distance, and for  ¥ t , it yields a constant value.It has a peak with high intensity at its center, where the velocity of the wave is less than its trailing edge.It has sturdy stability, which makes it vital for propagating over long region without substantial distortion.It can be used in signal processing, communicating over distant area, exploring properties of plasma waves, investigating soliton characteristics, and so on.The solution (4.5) revealed a bell-shaped soliton solution for the regarded values = a 5, l = -1.32,5, = y 1, = z 1, which is presented in figure 3. From figure 4, it is clear that, a bell-shaped soliton is established from solution (4.6) by picking out the values = a 5, l = -1.3Bell-shapedsolitons are distinguished solitons because of their extreme capacity to transfer data without any considerable attenuation of energy or shape.These

Figure 5
describes a periodic soliton solution of modulus of solution (4.7) for the apposite values = a 0soliton, which can propagate for a long time.Moreover, the solution (4.8) also characterizes periodic soliton for the estimated values = a 1.17, l = 1, shown in figure 6.
6.1.Effect of self-steeping and dispersive termsIn this subsection, we have visualized the effect of self-steeping and dispersive terms on solutions through 2D graphs of the obtained solutions drawn for a variety of values of a and c.The following figure 10 represents the diagrams of the absolute value of solution (4.4) for different values of a and c.The values of these parameters must be selected under constraints that the solution should not become imaginary or constant for the choosing values of parameters.

Table 2 .
Difference between solution of the mentioned methods.