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 generalized 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 expansion technique. The extended generalized expansion technique is a well-defined and simple technique which is based on the initial assumed solutions of the polynomial of . 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.
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], -expansion method [5], Kudryashov method [6], Lie symmetry method [7], extended trial equation method [8], singular manifold method [9], sinh–Gordon expansion method [10], -expansion method [11], generalized -expansion method [12], generalized-improved -expansion method [13], extended generalized -expansion method [14, 15], and expansion method [16].
Out of these methods, the extended generalized -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 with variable coefficients, and also, satisfies [14] where , , , and are constants. The Schamel, the Schamel Burgers [14], and modified Korteweg–De Vries (KdV) [15] equations are solved by the extended generalized –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] where , , and are functions of . 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 -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 -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] where , , , , and are smooth function of .
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 -symmetries, -reduction, and -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 -soliton-type traveling wave solutions; and Nakoulima et al. [35] studied the perturbation theory and the bifurcation theory.
In this paper, the extended generalized -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 Generalized Expansion Method
Considering the following partial differential equations, we have
Assume the initial solutions of Equation (4) given in [14] defined as where are functions of and satisfies Equation (1) and and . The general solution of (1) is given below in modified form with different situations:
Family-1. When and [15],
Family-2. When and [15],
Family-3. When and [15],
Family-4. When and [15],
Family-5. When and [15], where , , and , , , and are constants.
The steps of the algorithm are given below.
Step 1. Calculate the value of 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 and the value of . Then, substitute all the partial terms into Equation (4); then, Equation (4) becomes a polynomial of . Then, we have a system of differential equations by gathering the coefficients of the same power of term.
Step 3. Now, solve the obtained system for the values of , . Then, the required solutions of (4) can be obtained by substituting the values of in (5) and Equations (6)–(10).
3. Solution of the Generalized Zakharov–Kuznetsov Equation
Now, balancing between the term and the term of Equation (2), we get . Hence, the initially assumed solution of Equation (2) is of the form where , to be chosen in place of and , respectively, and satisfies Equation (1). Using the generalized wave transformation, , where , , and are functions of . Now, with the help of Equation (11), Equation (1), and the value of , we have the following results:
Now, substituting the values of Equations (11)–(16) in Equation (2), Equation (2) becomes a function of the polynomial of , then collects coefficients (equals zero) of the identical power of . The newly acquired system of equations is as follows:
Now solving Equations (17)–(24) together with the help of computational software Maple, the obtain solution of the system is as follows: where , , , , and are arbitrary constants and is the independent function of .
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: where the value of is the same as Equation (25) for Equation (26) and Equation (27), but the value of for Equation (28) and the value of for 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 term with the term in Equation (3), we get the value . For simplification, we choose , and the values of are chosen as , where , so the initial assumed solutions for Equation (3) are as follows:
Now, find the required partial derivative terms of Equation (33) with the help of Equation (1) and the value of ; then, substitute the values of partial derivative terms and Equation (33) into Equation (3). Then, Equation (3) becomes a function of the polynomial of ; then, gathering terms of the same power of the which is identical to zero, the acquired system of differential equations is as follows:
Now, solving Equations (34)–(39) together, we get the solution of the form where , , , and are constants and , and are arbitrary functions of .
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: where the value of is in (40) for both Equations (42) and (43).
Similarly, using (8), the rational solution of Gardner equation is of the form where
Similarly, again, using (9)–(10), the obtained hyperbolic and trigonometric solutions of the Gardner equation are as where 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 generalized expansion method using Maple/Mathematica and found that the solution allows shock/kink-type waves for and 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.
Figure 1 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.
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Similarly, Figure 9 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.
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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 , , are functions of , and satisfies the ordinary differential equation of the form , where and are constants. In our results, the solutions of the generalized ZK equation and the Gardner equation are of the form , , are functions of , and satisfies the ordinary differential equation of the form , where , , , and are constants.
6. Conclusion
In this study, we successfully used the extended generalized -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 expansion method. We concur that the system of nonlinear evolution equations might be solved using the method described above.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest, and all the authors agree to publish this paper under academic ethics.