# Exact Solutions of the Generalized ZK and Gardner Equations by Extended GeneralizedG′/G-Expansion Method

Authors: Sanjaya K. Mohanty; Apul N. Dev; Soubhagya Kumar Sahoo; Homan Emadifar; Geeta Arora
Journal: Advances in Mathematical Physics
 (2023)

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## Abstract

In this investigation, the exact solutions of variable coefficients of generalized Zakharov-Kuznetsov (ZK) equation and the Gardner equation are studied with the help of an extended generalizedG′/G expansion method. The main objective of this study is to establish the closed-form solutions and dynamics of analytical solutions to the generalized ZK equation and the Gardner equation. The generalized ZK equation and the Gardner equation govern the behavior of nonlinear wave phenomena in the presence of magnetic field in plasma dynamics, turbulence, bottom topography, and quantum field theory. We construct innovative solutions to the models under consideration using various computing tools and a recently developed extended generalized G′/G expansion technique. The extended generalized G′/G expansion technique is a well-defined and simple technique which is based on the initial assumed solutions of the polynomial of G′/G. The derived solutions for both the equations are the hyperbolic, trigonometric, and rational functions. The obtained solutions have shock/kink waves and multisoliton, which depict the dynamical representations of the acquired solutions through the three-dimensional surface plots and the contour plots.


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## Body

## 1. Introduction

The mathematical models are important as it impacts every aspects of our daily life (see [1]). The study of exact solutions of such mathematical models is the most exciting area of research investigation. To obtain the exact solutions of such mathematical models, recently, researchers follow different analytical methods, such as tangent hyperbolic method [2], modified extended direct algebraic method [3, 4], G′/G2-expansion method [5], Kudryashov method [6], Lie symmetry method [7], extended trial equation method [8], singular manifold method [9], sinh–Gordon expansion method [10], G′/G-expansion method [11], generalized G′/G-expansion method [12], generalized-improved G′/G-expansion method [13], extended generalized G′/G-expansion method [14, 15], and G′/G,1/G expansion method [16].Out of these methods, the extended generalizedG′/G-expansion method [14, 15] is a well-defined, simple, and effective method. It is based on initial assumption solution, which can be written in terms of polynomial of G′/G with variable coefficients, and also, G=Gβ satisfies [14]
(1)Q1GG′′−Q2GG′−Q3G′2−Q4G2=0,where Q1≠0, Q2, Q3, and Q4 are constants. The Schamel, the Schamel Burgers [14], and modified Korteweg–De Vries (KdV) [15] equations are solved by the extended generalized G′/G–expansion method and obtained the shock waves, periodic waves, and multiwaves.The generalized ZK equation with variable coefficients is written in the form of [17, 18]
(2)ut+atuux+btu2ux+uxxx+gtuxyy=0,where at, bt, and gt are functions of t. The generalized Zakharov–Kuznetsov equation with variable coefficients can be used to describe the travel of plasma in a magnetic field and the motion of water waves in 2+1-dimensional space [19, 20]. Yan and Liu [17] reduced the generalized ZK equation with variable coefficients into a simpler form using symmetries and obtained some new similarity solutions. Zayed and Abdelaziz [18] derived the unique polynomial solution to the generalized ZK equation, which has applications in solitary wave theory. Wakil et al. [21] used the exponential function approach to find solutions to the generalized ZK equation, where the solutions are in the form of the exponential function. Gao and Wang [22] solved the generalized ZK equation with variable coefficients using the unified technique, the modified Kudryashov method, and the F-expansion method. They got solitary, soliton, elliptic, rational, periodic, trigonometric, and hyperbolic trigonometric-type solutions.The Gardner equation with variable coefficient is an important model which is defined as [23]
(3)ut+atuux+btu2ux+htuxxx+gtux+ftu=0,where at, bt, ht, gt, and ft are smooth function of t.The Gardner equation is applied on many scientific fields such as bottom topography [24], magnetoplasma [25], plasma physics [26], mechanical analysis [27], and lattice Boltzmann model [28]. Ozkan and Yasar [29] studied the exact solutions of the nonlocal Gardner equation by employing the Hirota bilinear, extended homoclinic, and three-wave methods to obtain solutions of breather-type and multi-wave-type solitons. Orhan and Ozer [30] studied the μ-symmetries, μ-reduction, and μ-conservation laws for the Gardner equation. Cao and Du [31] used the trial equation method and obtained double-periodic-type soliton solutions for the variable coefficient Gardner equation. Singh and Gupta [32] used the Riccati equation mapping method and obtained solutions of kink soliton, periodic, and rational solution types. Abdou [33] used the bilinear method and extended homoclinic test approach to obtain breather-type soliton and two soliton solutions; Xu et al. [34] used the Hirota bilinear method to present one, two, and three-and N-soliton-type traveling wave solutions; and Nakoulima et al. [35] studied the perturbation theory and the bifurcation theory.In this paper, the extended generalizedG′/G-expansion method [14] is applied on variable coefficient generalized ZK equation (2) and the Gardner equation (3). The structure of the manuscript is as follows: the methodology of the expansion method is given in Section 2. Section 3 and Section 4 deal with solutions of the generalized ZK equation and Gardner equation with dynamic structures, respectively. We discuss the established results in Section 5 and finally present a brief conclusion.


## 2. Algorithm of the Extended GeneralizedG′/G Expansion Method

Considering the following partial differential equations, we have(4)Su,ux,ut,uxt,uxx,utt,⋯=0.Assume the initial solutions of Equation (4) given in [14] defined as
(5)ux,t=∑j=0mαjXG′Gm,αm≠0,where αjXj=0,1,⋯,m are functions of X and G=Gβ satisfies Equation (1) and β=βX and X=Xx,t. The general solution of (1) is given below in modified form with different situations:Family-1. WhenQ2≠0 and Q22+4Q4Q1−Q3>0 [15],
(6)G′G=Q22Ω+Δ2ΩD1sinhΔ/2Ωβ+D2coshΔ/2ΩβD1coshΔ/2Ωβ+D2sinhΔ/2Ωβ.Family-2. WhenQ2≠0 and Q22+4Q4Q1−Q3<0 [15],
(7)G′G=Q22Ω+−Δ2Ω−D1sin−Δ/2Ωβ+D2cos−Δ/2ΩβD1cos−Δ/2Ωβ+D2sin−Δ/2Ωβ.Family-3. WhenQ2≠0 and Q22+4Q4Q1−Q3=0 [15],
(8)G′G=Q22Ω+D2D1+D2β.Family-4. WhenQ2=0 and Q1−Q3Q4>0 [15],
(9)G′G=ΩQ4ΩD1sinhΩQ4/Ωβ+D2coshΩQ4/ΩβD1coshΩQ4/Ωβ+D2sinhΩQ4/Ωβ.Family-5. WhenQ2=0 and Q1−Q3Q4<0 [15],
(10)G′G=−ΩQ4Ω−D1sin−ΩQ4/Ωβ+D2cos−ΩQ4/ΩβD1cos−ΩQ4/Ωβ+D2sin−ΩQ4/Ωβ,where Ω=Q1−Q3, Δ=Q22+4Q4Ω, and D1, D2, D3, and D4 are constants.The steps of the algorithm are given below.Step 1.
Calculate the value ofm by balancing the terms of highest order derivatives and with nonlinear terms of Equation (4) with the help of Equation (1) and Equation (5).Step 2.
Then, evaluate the partial derivative terms of Equation (5) with the help of the function β and the value of m. Then, substitute all the partial terms into Equation (4); then, Equation (4) becomes a polynomial of G′/G. Then, we have a system of differential equations by gathering the coefficients of the same power of G′/G term.Step 3.
Now, solve the obtained system for the values ofαjX, j=0,1,2,⋯,m. Then, the required solutions of (4) can be obtained by substituting the values of αjX in (5) and Equations (6)–(10).


## 3. Solution of the Generalized Zakharov–Kuznetsov Equation

Now, balancing between the termuxxx and the term u2ux of Equation (2), we get m=1. Hence, the initially assumed solution of Equation (2) is of the form
(11)ux,y,t=α0t+α1tG′G,where α0t, α1t≠0 to be chosen in place of α0X and α1X, respectively, and G=Gβ satisfies Equation (1). Using the generalized wave transformation, ux,y,t=uβ,β=ptx+qty+rt, where pt, qt, and rt are functions of t. Now, with the help of Equation (11), Equation (1), and the value of β, we have the following results:
(12)ut=α1txdptdt+ydqtdt+drtdtQ3Q1−1G′G2+dα1tdt+α1txdptdt+ydqtdt+drtdtQ2Q1G′G+α1txdptdt+ydqtdt+drtdtQ4Q1+dα0tdt,(13)ux=ptα1tQ3Q1−1G′G2+ptα1tQ2Q1G′G+ptα1tQ4Q1,(14)uxxx=6p3tα1tQ3Q1−13G′G4+12p3tα1tQ3Q1−12Q2Q1G′G3+8p3tα1tQ3Q1−12Q4Q1+7p3tα1tQ3Q1−1Q22Q12G′G2+8p3tα1tQ3Q1−1Q2P4Q12+p3tα1tQ23Q13G′G+2p3tα1tQ3Q1−1Q42Q12+p3tα1tQ22Q4Q13,(15)u2ux=ptα13tQ3Q1−1G′G4+2ptα0tα12tQ3Q1−1+ptα13tQ2Q1G′G3+ptα13tQ4Q1+2ptα0tα12tQ2Q1+ptα02tα1tQ3Q1−1G′G2+ptα02tα1tQ2Q1+2α0tα12tptP4P1G′G+ptα02tα1tQ4Q1,(16)uxyy=6ptq2tα1tQ3Q1−13G′G4+12ptq2tα1tQ3Q1−12Q2Q1G′G3r+8ptq2tα1tP3Q1−12Q4Q1+7ptq2tα1tQ3Q1−1Q22Q12G′G2+8ptq2tα1tQ3Q1−1Q2Q4Q12+ptq2tα1tQ23Q13G′G+2ptq2tα1tQ3Q1−1Q42Q12+ptq2tα1tQ22Q4Q13.Now, substituting the values of Equations (11)–(16) in Equation (2), Equation (2) becomes a function of the polynomial of G′/G, then collects coefficients (equals zero) of the identical power of G′/G. The newly acquired system of equations is as follows:
(17)G′G4:btα13tptQ3Q1−1+6α1tp3tQ3Q1−13+6gtα1tptq2tQ3Q1−13=0,(18)G′G3:atα12tptQ3Q1−1+btα13tptQ2Q1+2btα12tptα0tQ3Q1−1+12α1tp3tQ3Q1−12Q2Q1+12gtα1tptq2tQ3Q1−13Q2Q1=0,(19)G′G2:α1tdrtdtQ3Q1−1+atα0tα1tptQ3Q1−1+atα12tptQ2Q1+btα13tptQ4Q1+btα02tα1tptQ3Q1−1+2btα12tptα0tP2P1+8α1tp3tQ3Q1−12P4P1+7α1tp3tQ3Q1−1Q22Q12r+8gtα1tptq2tQ3Q1−12Q4Q1+7gtα1tptq2tQ3Q1−1Q22Q12=0,(20)G′G1:atα12tptQ4Q1+atα0tα1tptQ2Q1+2btα12tα0tptQ4Q1+btα02tα1tptQ2Q1+8α1tp3tQ3Q1−1Q2Q4Q12+α1tp3tQ23Q13+8gtα1tptq2tQ3Q1−1Q2Q4Q12+gtα1tptq2tQ23Q13+dα1tdt+α1tdrtdtQ2Q1=0,(21)G′G0:α1tdrtdtQ4Q1+dα0tdt+atα0tα1tptQ4Q1+btα02tα1tptQ4Q1+2α1tp3tQ3Q1−1Q42Q12+α1tp3tQ22Q12Q4Q1+2gtα1tptq2tQ3Q1−1Q42Q12+gtα1tptq2tQ22Q12Q4Q1=0,(22)xG′G2:α1tdptdtQ3Q1−1=0,yG′G2:α1tdqtdtQ3Q1−1=0,(23)xG′G1:α1tdptdtQ2Q1=0,yG′G1:α1tdqtdtQ2Q1=0,(24)xG′G0:α1tdptdtQ4Q1=0,yG′G0:α1tdqtdtQ4Q1=0.Now solving Equations (17)–(24) together with the help of computational software Maple, the obtain solution of the system is as follows:
(25)α1t=s5,pt=s4,qt=s3,α0t=s2,gt=−bts52Q12−6s42Ω26s32Ω2,at=−−2btΩs2+1/2s5Q2Ω,rt=−1Ω2s4Ω2s22+s5Ωs2−13Q4Ω−12Q22s52∫btdt+s1,where s1, s2, s3, s4, and s5 are arbitrary constants and bt is the independent function of t.Now, substituting the values of Equations (6)–(10) separately and Equation (25) in Equation (11), we have hyperbolic, trigonometric, rational, hyperbolic, and trigonometric solutions of generalized ZK equation, respectively, as follows:
(26)ux,y,t=s2+s5Q22Ω+s5Δ2ΩD1sinhΔ/2Ωs4x+s3y+rt+D2coshΔ/2Ωs4x+s3y+rtD1coshΔ/2Ωs4x+s3y+rt+D2sinhΔ/2Ωs4x+s3y+rt,(27)ux,y,t=s2+s5Q22Ω+s5−Δ2Ω−D1sin−Δ/2Ωs4x+s3y+rt+D2cos−Δ/2Ωs4x+s3y+rtD1cos−Δ/2Ωs4x+s3y+rt+D2sin−Δ/2Ωs4x+s3y+rt,(28)ux,y,t=s2+s5Q22Ω+C2D1+D2s4x+s3y+rt,(29)ux,y,t=s2+s5ΩQ4ΩD1sinhΩQ4/Ωs4x+s3y+rt+D2coshΩQ4/Ωs4x+s3y+rtD1coshΩQ4/Ωs4x+s3y+rt+D2sinhΩQ4/Ωs4x+s3y+rt,(30)ux,y,t=s2+s5−ΩQ4Ω−D1sin−ΩQ4/Ωs4x+s3y+rt+D2cos−ΩQ4/Ωs4x+s3y+rtD1cos−ΩQ4/Ωs4x+s3y+rt+D2sin−ΩQ4/Ωs4x+s3y+rt,where the value of rt is the same as Equation (25) for Equation (26) and Equation (27), but the value of
(31)rt=−16Q42s4Q22s2216Q42−s54Q4−5s5212Q4∫btdt+s1for Equation (28) and the value of
(32)rt=−1Ω2s4Ω2s22+s5Ωs2−13Q4Ωs52∫btdt+s1for Equations (29) and (30).The graphical representation of the acquired solutions of the generalized ZK equation with variable coefficients is demonstrated through the computational mathematical software with the precise values of the free parameters.


## 4. Solutions of the Gardner Equation

Now, balancing the termuxxx with the term u2ux in Equation (3), we get the value m=1. For simplification, we choose β=ptx+qt, and the values of αjX′ are chosen as αjt′, where j=0,1, so the initial assumed solutions for Equation (3) are as follows:
(33)ux,t=α0t+α1tG′G.Now, find the required partial derivative terms of Equation (33) with the help of Equation (1) and the value of β; then, substitute the values of partial derivative terms and Equation (33) into Equation (3). Then, Equation (3) becomes a function of the polynomial of G′/G; then, gathering terms of the same power of the G′/G which is identical to zero, the acquired system of differential equations is as follows:
(34)G′G4:btptα13tQ3Q1−1+6p3tα1thtQ3Q1−13=0,(35)G′G3:atα12tptQ3Q1−1+btptα13tQ2Q1+2ptα0tα12tbtQ3Q1−1+6p3tα1thtQ3Q1−121+Q2Q1=0,(36)G′G2:α1tdqtdtQ3Q1−1+atα0tptα1tQ3Q1−1+atα12tptQ2Q1+btptα02tα1tQ3Q1−1+ptbtα13tP4P1+2ptα0tα12tbtQ2Q1+8p3tα1thtQ3Q1−12Q4Q1+7p3tα1thtP22P12Q3Q1−1+ptα1tgtQ3Q1−1=0,(37)G′G1:dα1tdt+α1tdqtdtQ2Q1+ptatα0tα1tQ2Q1+ptatα12tQ4Q1+ptbtα02tα1tQ2Q1+2ptbtα0tα12tQ4Q1+8p3thtα1tQ3Q1−1Q2Q4Q12+p3tα1thtQ23Q13+ptα1tgtQ2Q1+ftα1t=0,(38)G′G0:α1tdqtdtQ4Q1+dα0tdt+atα0tα1tptQ4Q1+ptbtα02tα1tQ4Q1+2p3tα1thtQ3Q1−1Q42Q12+p3tα1tQ22Q4Q13+ptα1tgtQ4Q1+ftα0t=0,(39)xG′G2:α1tdptdtQ3Q1−1=0,xG′G1:α1tdptdtQ2Q1=0,xG′G0:α1tdptdtQ4Q1=0.Now, solving Equations (34)–(39) together, we get the solution of the form
(40)pt=s4,α0t=s2e∫−ftdt,α1t=s3e∫−ftdt,at=6s42htΩ2s2Ω+s3Q2Q12s32e∫ftdt,(41)bt=−6s42htΩ2Q12s32e∫2ftdt,qt=−s4Q12s32∫s42htQ22s32−2Q4s32Ω+6Q2s2Ω+6s22Ω2+s32gtQ12dt+s1,where s1, s2, s3, and s4 are constants and ft,ht, and gt are arbitrary functions of t.Now, putting the value of Equation (40) and Equation (6)–Equation (7) into Equation (33), then the obtain hyperbolic function solutions and trigonometric function solutions for the Gardner equation are as follows:
(42)ux,t=s2e∫−ftdt+s3e∫−ftdt⋅Q22Ω+Δ2ΩD1sinhΔ/2Ωs4x+qt+D2coshΔ/2Ωs4x+qtD1coshΔ/2Ωs4x+qt+D2sinhΔ/2Ωs4x+qt,(43)ux,t=s2e∫−ftdt+s3e∫−ftdtQ22Ω++s3e∫−ftdt−Δ2Ω⋅−D1sin−Δ/2Ωs4x+qt+D2cos−Δ/2Ωs4x+qtD1cos−Δ/2Ωs4x+qt+D2sin−Δ/2Ωs4x+qt,where the value of qt is in (40) for both Equations (42) and (43).Similarly, using (8), the rational solution of Gardner equation is of the form
(44)ux,t=s2e∫−ftdt+s3e∫−ftdtQ22Ω+D2D1+D2s4x+qt,where
(45)qt=−s4Q12s32∫s42ht32Q22s32+6Q2s2Ω+6s22Ω2+s32gtQ12dt+s1.Similarly, again, using (9)–(10), the obtained hyperbolic and trigonometric solutions of the Gardner equation are as
(46)ux,t=s2e∫−ftdt+s3e∫−ftdtΩQ4ΩD1sinhΩP4/Ωs4x+qt+D2coshΩP4/Ωs4x+qtD1coshΩP4/Ωs4x+qt+D2sinhΩP4/Ωs4x+qt,(47)ux,t=s2e∫−ftdt+s3e∫−ftdt−ΩQ4Ω−D1sin−ΩP4/Ωs4x+qt+D2cos−ΩP4/Ωs4x+qtD1cos−ΩP4/Ωs4x+qt+D2sin−ΩP4/Ωs4x+qt,where
(48)qt=−s4Q12s32∫s42ht6s22Ω2−2Q4s32Ω+s32gtQ12dt+s1,for both (46) and (47).The graphical representation of the obtained solutions of the Gardner equation with variable coefficients is demonstrated through the three-dimensional surface and contour plots.


## 5. Results with Analysis

We have studied the extended generalizedG′/G expansion method using Maple/Mathematica and found that the solution allows shock/kink-type waves for Q22+4Q4Q1−Q3>0 and Q4Q1−Q3>0 to the generalized ZK equation and the Gardner equation with variable coefficients. While accounting for multiple random values for the free parameters, we attempted to produce a numerical representation of each solution in time and space. The solutions contain the tangential hyperbolic term, which permits shock waves.The details of the obtained results for the generalized ZK equation and the Gardner equation with variable coefficients are demonstrated through the three-dimensional and contour plots. For the solution for the generalized ZK equation with variable coefficients, Equation (26) and Equation (29) are of hyperbolic function type, Equation (27) and Equation (30) are of trigonometric function, and Equation (28) is of the rational type.Figure1 represents the shock wave profile for Equation (26), and the transition of the wave is observed through the contour plots. Similarly, Figure 2 represents the shock wave for Equation (29). Figure 3 demonstrates the multisoliton wave profile for Equation (27). Similarly, Figure 4 demonstrates the multisoliton of Equation (28). Figure 5 shows the multisoliton structures with periodic, and the contour plot shows the partition of the plots. Figure 6 demonstrates the typical three-dimensional surface plot and contour plot of Equation (42), which represents a shock wave profile, where the shock transition is observed through the contour plot. Similarly, Figure 7 shows the shock wave profile for Equation (46). Figure 8 represents the three-dimensional surface plot of Equation (43) with space-time coordinates in which the multisoliton structures are obtained, where the dark and bright solitons can be observed from the corresponding contour plot.Figure 1
(a) Shows the three-dimensional surface plot of (26), and (b) shows the contour plot of (26), where Q1=30, Q2=1, Q3=2, Q4=1, D1=1, D2=0, s1=2, s2=3, s3=1, s4=2, s5=2, bt=t, y=1, −20≤x≤20, and −1<t≤1.
(a)(b)Figure 2
(a) Shows the is three-dimensional surface plot of (29), and (b) shows the contour plot of (29), where Q1=15, Q2=0, Q3=2, Q4=1, D1=1, D2=0, s1=2, s2=3, s3=1, s4=2, s5=2, bt=t, y=1, −20≤x≤20, and −10<t≤10.
(a)(b)Figure 3
(a) Shows the three-dimensional surface plot of (27), and (b) shows the contour plot of (27), where Q1=20, Q2=1, Q3=2, Q4=−1, D1=1, D2=0, s1=2, s2=3, s3=1, s4=2, s5=2, y=1, bt=t, −5≤x≤5, and −1<t≤1.
(a)(b)Figure 4
(a) Shows the three-dimensional surface plot of (28), and (b) shows the contour plot of (28), where Q1=3, Q2=4, Q3=5, Q4=2, D1=1, D2=1, s1=2, s2=3, s3=1, s4=2, s5=2, bt=t, y=1, −5≤x≤5, and −1<t≤1.
(a)(b)Figure 5
(a) Shows the three-dimensional surface plot of (30), and (b) shows the contour plot of (30), where Q1=4, Q2=0, Q3=2, Q4=−1, D1=1, D2=0, s1=2, s2=3, s3=1, s4=2, s5=2, bt=t, y=1, −10≤x≤10, and −10<t≤10.
(a)(b)Figure 6
(a) Shows three-dimensional surface plot of Equation (42), and (b) shows the contour plot of Equation (42), where Q1=30, Q2=1, Q3=2, Q4=1, s1=2, s2=3, s3=1, s4=2, D1=1, D2=0, ft=t2, gt=t2, and ht=t2, within −10≤x≤10 and −0.1≤t≤0.1.
(a)(b)Figure 7
(a) Shows the three-dimensional surface plot of Equation (46), and (b) shows the contour plot of Equation (46), where Q1=40, Q2=0, Q3=2, Q4=1, s1=2, s2=3, s3=1, s4=2, D1=1, D2=0, ft=t2, gt=t2, and ht=t2, within −10≤x≤10 and −0.1≤t≤0.1.
(a)(b)Figure 8
(a) Shows the three-dimensional surface plot of Equation (43) and (b) shows the contour plot of Equation (43), where Q1=13, Q2=1, Q3=2, Q4=−1, s1=2, s2=3, s3=1, s4=2, D1=1, D2=0, ft=t2, gt=t2, and ht=t2, within −5≤x≤5 and −0.1≤t≤0.1.
(a)(b)Similarly, Figure9 represents the three-dimensional surface plot of Equation (44) and represents the multisoliton structures. Figure 10 illustrates the three-dimensional surface plot of Equation (47), which shows the multisoliton and periodic structures, where the repetition of the graphs can be observed from the corresponding contour plots.Figure 9
(a) Shows the three-dimensional surface plot of Equation (44) and (b) shows the contour plot of Equation (44), where Q1=3, Q2=4, Q3=5, Q4=2, s1=2, s2=3, s3=1, s4=2, D1=1, D2=1, ft=t2, gt=t2, and ht=t2, within −5≤x≤5 and −1≤t≤1.
(a)(b)Figure 10
(a) Shows the three-dimensional surface plot of Equation (47) and (b) shows the contour plot of Equation (47), where Q1=3, Q2=0, Q3=2, Q4=−1, s1=2, s2=3, s3=1, s4=2, D1=1, D2=0, ft=t2, gt=t2, and ht=t2, within −5≤x≤5 and −0.1≤t≤0.1.
(a)(b)We have tried to validate our results over mathematical and analytical findings with the available existing literature. Zayed and Abdelaziz investigated that the solutions for both ZK [18] and the Gardner equation [17] with variable coefficients are of the form u=α0t+α1tG′/G, α0, α1≠0 are functions of t, and G=Gβ satisfies the ordinary differential equation of the form G′′+Q1G′+Q2G=0, where Q1 and Q2 are constants. In our results, the solutions of the generalized ZK equation and the Gardner equation are of the form u=α0t+α1tG′/G, α0, α1≠0 are functions of t, and G=Gβ satisfies the ordinary differential equation of the form Q1GG′′−Q2GG′−Q3G′2−Q4G2=0, where Q1≠0, Q2, Q3, and Q4 are constants.


## 6. Conclusion

In this study, we successfully used the extended generalizedG′/G-expansion technique to find exact solutions to the generalized ZK equation and the Gardner equation with variable coefficients. The derived solutions under consideration have shock/kink waves and multisoliton waves as their dynamical structures. The approach taken and the solutions discovered could aid in understanding the propagation properties of solitary and shock/kink waves. These results can also be used in a degenerate plasma medium, which is frequently studied in optical fibers [36], etc. In addition to practical uses, numerical solvers can compare the accuracy of their results and help with stability analysis by using closed-form solutions of nonlinear evolution equations. Any nonlinear evolution problem with a known balancing property can be solved using the extended generalized G′/G expansion method. We concur that the system of nonlinear evolution equations might be solved using the method described above.



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