Noval soliton solution, sensitivity and stability analysis to the fractional gKdV-ZK equation

This work examines the fractional generalized Korteweg-de-Vries-Zakharov-Kuznetsov equation (gKdV-ZKe) by utilizing three well-known analytical methods, the modified \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \frac{G^{'}}{G^2}\right)$$\end{document}G′G2-expansion method, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \frac{1}{G^{'}}\right)$$\end{document}1G′-expansion method and the Kudryashov method. The gKdV-ZK equation is a nonlinear model describing the influence of magnetic field on weak ion-acoustic waves in plasma made up of cool and hot electrons. The kink, singular, anti-kink, periodic, and bright soliton solutions are observed. The effect of the fractional parameter on wave shapes have been analyzed by displaying various graphs for fractional-order values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}β. In addition, we utilize the Hamiltonian property to observe the stability of the attained solution and Galilean transformation for sensitivity analysis. The suggested methods can also be utilized to evaluate the nonlinear models that are being developed in a variety of scientific and technological fields, such as plasma physics. Findings show the effectiveness simplicity, and generalizability of the chosen computational approach, even when applied to complex models.

where a,b, and d are the constants.The gKdV-ZK fractional equation is a special type of nonlinear evolution equation that can be used to describe different complex nonlinear phenomena in the various fields of nonlinear science such as , plasma physics, fluid dynamics, and electromagnetism.The analytical solutions of (1) were attained by utilizing Kudryashov's technique, and Jacobi elliptic function scheme 36 .The hot isothermal and warm adiabatic fluid mixtures were derived in 37 .The electron acoustic solitons for a small amplitude region were investigated in 38 .The exact solutions of Eq. (1) were attained by utilizing Kudryashov's technique, and Jacobi elliptic function scheme 36 .The kink, quasi-periodic and lump-type soliton of Eq. (1) were acquired by utilizing the Lie symmetry approuch 39 .In the past modified G ′ G 2 -expansion technique, 1 G ′ -expansion approach and the Kudryashov scheme were used on different equation such as: In 40 the variety of traveling solution was obtained.In 41 , the analytical solutions for Gardner equations were achieved by utilizing 1 G ′ -expansion technique.By utilizing the modified G ′ G 2 -expansion approach, the traveling wave solutions were obtained for the nonlinear Schrodinger equation in 42 .The soliton solutions of the Fokas-Lenells model also have been attained by utilizing G ′ G 2 -expansion approach 43 .The topological, periodic, and singular soliton solutions were attained in 44 by utilizing the Kudryashov method.The soliton solutions of the Maccari equation were investigated with the aid of the Kudryashov scheme 45 .Different definitions for fractional derivatives have been utilized in the last many years.Such as, Beta time-fractional 46 , Reimann-Liouville 47 , Caputo fractional 48 , Conformable fractional 49 , truncated M-fractional derivative 50 .
This research work is divided into sections: In section(2) we described the Beta derivative.In section(3) modified G ′ G 2 -expansion method is utilized on Eq. ( 2) to attained the periodic and singular type soliton .The kink and dark type soliton are retrieved by using 1 G ′ -expansion method in section(4).Section (5) discussed the Kudryashov scheme.The sensitivity and stability analysis of the soliton solution is discussed in section (6).In section (7) graphically representation.In the end, the conclusion is presented in section (8).

Beta derivative
Definition: Let P(t) be a function defined ∀ non-negative t.Then, the β derivative of P(t) of order β is given by 51

The modified G ′ G 2 -expansion method
Consider the NPDE is where operator D represents the partial derivative and u is an unknown function.
Consider the travelling wave is utilizing (3) into (2),then The travelling wave solutions are (1) www.nature.com/scientificreports/where τ 1 , σ 1 and b n are unknown parameters which find latter.The (6) has three cases: where A 1 and B 1 are arbitrary nonzero constants.
To obtain the three types of solution by putting the value of unknown b n and Eqs. ( 7),( 8),( 9) into (5).

Application of modified G ′ G 2 -expansion method
The gKdV-ZKe equation is, Suppose the transformation, on (10), we get Integrate (12) one time with respect to η , we get Utilizing the homogenous balance approach on (10), then we have N = 1, Utilizing ( 14) into ( 13), then we get, : The solution of the above system is given below, Vol:.( 1234567890)

Set-2
Equation ( 14) become, Three different solutions are given below, The 1

G ′ -expansion method
Consider the Eqs.( 2), ( 3), ( 4).The solution of ( 4) is, The second order ODE is, Vol www.nature.com/scientificreports/where a n , σ 1 and τ 1 are unknown parameters to be determined later and N is homogenous balance number.The (26) become, Then, Here, A 1 and A 2 are unknown parameters.Putting ( 25) into (4) and utilizing (26), then (4) can be changed into a polynomials of ( 1 G ′ ) .After this, we are setting the polynomial equal to zero, and then we get a system of algebraic equations.Solving the obtained system with the aid of Mathematica to attain the values of parameters.

Application of 1 G ′ -expansion method
Utilizing N = 1 into (25), then we have utilizing Eq. ( 29) into the Eq. ( 13) then we get set of algebraic equations Solving the overhead system of the equation we acquire the solutions,

Kudryashov method
Solution of Eq. ( 4) is, where b i is unknown, N is homogenous balance number, and Q(η) is the solution, 4κ 2 e γ η +ρe −γ η , Now putting (34) into (12) and obtaining the algebraic system by solving the system we lead soliton solution of the NPDE Eq. (1).

Application of Kudryashov method
Substituting N = 1 into (34) then, ( 26) Resolving the above system of equations we get the following solutions,

Sensitivity analysis
From (11), we can write as Let c b+2d = A and a b+2d = B then we get, Using the Galilean transformation on (41) then we get dynamical system as: We will now investigate the sensitive phenomena of the perturbed system shown below.Subsequently, we will decompose the schemes given in Eq. ( 42) into an autonomous conservative dynamical system (ACDS), as illustrated below: In which f represents to be the frequency and m 0 is the strength of the perturbed component 52 .In the current part of the investigation, we will explore whether the frequency term has any effect on the model which will be examined.To do this, we will evaluate the model under examination's particular appearance and address the impact of the perturbation's force and frequency.By using four different beginning conditions in the component, we aim to evaluate the sensitivity of such a solution to the perturbed dynamical structural Eq. ( 43) at the value of parameters c = 0.05, a = 0.5, b = d = f = 0.2, m 0 = 4.5.From Fig. 1 we have seen that In Fig(a), the system is not sensitive because there is overlapping in the cure but with a small change in the initial condition system becomes sensitive.

Stability analysis
The stability of the solitary wave solution is discussed in this section with the help of the Hamiltonian system.The HSM condition is given by 53 , Here, U represent the dependent variable, a 1 , a 2 are arbitrary constants and satisfies a 1 < a 2 .The following criteria determine how dependent the stability of the obtained solutions is on the HSM: where c is the sp eed of waves.The selected values for parameter is given by (g 1 = 0.1, ν = −0.8,g 3 = 0.3, τ = 0.05, y = 0.5, g 2 = 0.08, z = 0.5, ς = 0.1) make the ( 33) and ( 37 (

Results and discussion
This section discusses the graphical presentation of the gKdV-ZK equation.The physical phenomena of the nonlinear model are determined by giving suitable values to the arbitrary constants with the help of Mathematica.We illustrate 2 and 3-dimensional Figs.graph with respect to time t is presented in Fig. 4d.We have also observed that the solitary waves tiny shifts when the change fractional order beta is without changing the shape of the curve.Furthermore, we have compared our solutions with Romana et al. 54 that have attained bright and single soliton forms with the aid of an improved modified extended tanh expansion method (METEM).But in this article, we have achieved different forms   Figure Effect of parameter β on (33).

Figure
Figure Effect of parameter β on(18).