Traveling wave solutions for (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity

Abstract In this work, we consider the (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity. Solitary wave solutions, soliton wave solutions, elliptic wave solutions, and periodic (hyperbolic) wave rational solutions are obtained by means of the unified method. The solutions showed that this method provides us with a powerful mathematical tool for solving nonlinear conformable fractional evolution equations in various fields of applied sciences.


Introduction
Nonlinear fractional partial di erential equations (FPDEs) show a rich variety of nonlinear phenomena, arise in many physical and engineering applications like geophysical uid mechanics, uid mechanics, plasma physics, superconductivity, and optics. Therefore, seeking exact solutions of nonlinear FPDEs speci cally the nonlinear fractional evolution equations (NLFEEs) plays an important duty in the study of nonlinear physical phenomena [1][2][3][4][5][6][7][8][9]. It is signi cant to nd new solutions, since either new exact solutions or numerical approximate solutions may provide more information for understanding the physical phenomena. For an overview and recent developments of the local approach to fractional calculus we refer the reader to [10,11] and references therein.
Very recently, Khalil et al. [12] suggested a conformable fractional derivative. The new fractional derivative is very interesting and is getting an increasing of interest [13][14][15][16][17]. The conformable fractional derivatives didn't have a physical meaning as the Caputo or Riemann-Liouville derivatives. This situation is a general open problem for fractional calculus. Despite this many physical applications of conformable fractional derivative appear in the literature. Dazhi Zhao and Maokang Luo generalized the conformable fractional derivative and give the physical interpretation of generalized conformable derivative [18]. In addition, with the help of this fractional derivative and some important formulas, one can convert conformable fractional partial di erential equations into integer-order di erential equations by traveling wave transformation [19]. Later on, many researchers established exact traveling wave solutions of various nonlinear fractional evolution equations via this fractional derivative. For example, Eslami [20] solved nonlinear fractional coupled nonlinear Schrodinger equations by using the Kudryashov method. Kaplan [21] proposed the modi ed simple equation method and the exponential rational function method to solve the nonlinear conformable time-fractional Boussinesq equation. Korkmaz [22] applied modi ed Kudryashov method to obtain the exact solutions of the the (3+1) conformable time-fractional Jimbo-Miwa, Zakharov-Kuznetsov and Modi ed Zakharov-Kuznetsov equations. Aminikhah et al. [23] used the sub equation method to obtain the exact solutions of the fractional (1+1) and (2+1) regularized long-wave equations which arise in several physical applications, including ion sound waves in plasma. Rezazadeh et al. [24,25] concerned about the same method for obtaining traveling wave solutions for the conformable fractional generalized Kuramoto-Sivashinsky equation and fractional Zakharov-Kuznetsov equation with dual-power law nonlinearity. Tariq et al. [26] investigated the new exact solutions of a nonlinear evolution equation that appear in mathematical physics, speci cally Cahn-Allen equation by applying tanh method. Akbulut et al. [27] obtained exact solutions for (2+1)-dimensional time-fractional Zoomeron equation and the time-fractional third order modi ed KdV equation via the auxiliary equation method. Ekici et al. [28] proposed the rst integral method to study the optical solitons with fractional temporal evolution in presence of Hamiltonian perturbation terms governed by three types of nonlinearity. Cenesiz et al. [29] obtained some exact solutions for time-fractional Burgers equation, modi ed Burgers equation and Burgers-Korteweg-de Vries equation via the same method. Kurt et al. [30] established some traveling wave solutions for fractional Nizhnik-Novikov-Veselov and fractional Klein-Gordon equations via the Exp-Function Method. Tasbozan et al., [31] solved nonlinear fractional Boussinesq and combined KdV-mKdV equations by using Jacobi elliptic function expansion method. Eslami et al. [32,33] proposed the rst integral method and functional variable method to solve the space-time fractional Schrödinger-Hirota equation and the spacetime fractional modi ed KDV-Zakharov-Kuznetsov equation and fractional Bogoyavlenskii equations, respectively. Hosseini et al. [34] used the ansatz method to obtain the exact solutions of the fractional Klein-Gordon equations with quadratic and cubic nonlinearities. Eslami [35] applied G /G−expansion method to obtain the exact solutions of the space-time fractional (2+1)-dimensional dispersive long wave equations. Cenesiz et al. [36] applied the functional variable method to obtain the exact solutions of fractional modi ed KdV-ZK equation and Maccari system. Kaplan et al. [37] solved (2+1)-dimensional conformable time-fractional Zoomeron equation and the conformable space-time fractional EW equation by using modi ed simple equation method.
This paper is organized as follows: In section 2, we recall some basic de nitions of the conformable fractional derivative. In section 3, the key idea of our method is de-scribed. Sections 4 is devoted to the application of the UM for (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation. Conclusions are outlined in section 5.

Conformable fractional derivative
Here, we introduce some basic properties and de nitions of the conformable fractional calculus theory which can be found in [12,13]. De nition 2.1 Let f : ( , ∞) → R, then, the conformable fractional derivative of f of order α is de ned as [12] t D α f (t) = lim The new de nition satis es the properties which given in the following theorem. Theorem 1 Let α ∈ ( , ], and f , g be α-di erentiable at a point t, then [12] ( In addition, if f is di erentiable, then t D α f (t) = t −α df dt . In [13], T. Abdeljawad established the chain rule for conformable fractional derivatives as in the following theorem. Theorem 2 Let f : ( , ∞) → R be a function such that f is di erentiable and also α-di erentiable. Let g be a function de ned in the range of f and also di erentiable; then, one has the following rule

The description of the UM
In this section we describe the UM for nding exact solutions of nonlinear conformable fractional evolution equations.
Consider the following nonlinear conformable fractional evolution equation in two variables and a dependent variable u as where F is a polynomial in its arguments in which the highest order derivatives and nonlinear terms are involved. To solve Eq. (4), we take the traveling wave transformation This enables us to use the following changes Substituting Eq. (5) into Eq. (4) yields a nonlinear ordinary di erential equation as following where where p i , c i , α , and αs are arbitrary constants to be determined later. It is worth to be noticing that, n and k are determined from the balance equation by the criteria given in [39][40][41][42][43][44][45]. Also, a second condition (the consistency condition), which asserts that the arbitrary functions in Eq. (7) could be consistently determined, is used. When p = , (7) solves to elementary solutions (explicit or implicit) while when p = , it solves to elliptic solutions.

. The rational function solution
To get the rational function solutions of Eq.
where p i , q i , c i , α , and αs are arbitrary constants to be determined later. It is worth to be noticing that, n, r and k are determined from the balance equation by the criteria given in [39][40][41][42][43][44][45]. Also, a second condition (the consistency condition), which asserts that the arbitrary functions in Eq. (8) could be consistently determined, is used. When p = , (8) solves to elementary solutions (explicit or implicit) while when p = , it solves to elliptic solutions.

The (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity
In Eq.(1), a and b are real valued constants. The rst term is the evolution term, while the coe cients of a and b respectively, are the nonlinearity and dispersion. Also the parameter n is the power law nonlinearity parameter. Solitons are the result of a delicate balance between dispersion and nonlinearity. Eq.(1) typically appears in the study of plasma physics. B. T. Matebese et al. [52] solved the (3+1) dimensional Zakharov-Kuzetsov equation by G /G−expansion method, extended tanh-function method and ansatz metod. Furthermore, Aminikhah et al. [53] proposed the functional variable method to solve this equation when α = . The special case where n = and α = gives the (3+1) dimensional Zakharov-Kuzetsov equation [54]. Let with puting the relation (9) and its derivatives in to the Eq.(1) −λU ξ + aU n U ξ + b(U ξξ + U ξξ + U ξξ ) ξ = .

The polynomial solutions
To nd the polynomial solutions of the fractional (3+1)-Zakharov-Kuznetsov equation with power law nonlinearity, we assume that where p i and b i are arbitrary constants. By considering the homogeneous balance between V V ξξ and V in Eq. (12), we get n = (k − ), k = , , , . . . . Here, we con ne ourselves to nd these solutions when k = and p = or p = . So, we suppose that the polynomial solution of the ODE (12) has the form .

. The solitary wave solution
To obtain these solutions, we put p = in the auxiliary equation given by (14). From Eq. (14) when p = , we have When we use Eq.(15) into Eq. (12) and equating the coecients of ϕ(ξ ) to zero we get a system of algebraic equations. By solving this algebraic system of equations with the help of MATHEMATICA or MAPLE, it yields the following: where By solving the auxiliary equation ϕ (ξ ) = b +b ϕ(ξ )+ b ϕ (ξ ) and substituting together with (16) into Eq.(12), we get the solution of Eq.(1) namely n α t α and < α ≤ . Fig. 1 depicts the 3D and 2D charts of the solution given by u (x, y, z, t) with the parameters a = . , b = . , n = , and R = . .

. The soliton wave solution
Here, we put p = in the auxiliary equation given by (14). From Eq. (14) when p = , we have By substituting from (18) into Eq.(12) and by a similar way as we did in the last case, we get By solving the auxiliary equation ϕ (ξ ) = ϕ(ξ ) b + b ϕ(ξ ) + b ϕ (ξ ) and substituting together with (19) into Eq.(18), we get the solution of eq.(1) namely b α t α and < α ≤ . Fig. 2 depicts the 3D and 2D charts of the solution given by u (x, y, z, t) with the parameters a = . , b = − . , b = , and b = .

. . The elliptic wave solution
In this section we nd the complex elliptic wave solution.

. The rational solutions
To nd the rational solutions of the fractional (3+1)-Zakharov-Kuznetsov equation with power law nonlinearity, we assume that where p i , q i and b i are arbitrary constants. By considering the homogeneous balance between V V and V in Eq. (12), we get n − r = (k − ), k = , , , . . . .
Here, we nd these solutions when k = (so n = r) and p = . So from (25), we have two cases as follow

. . Case 1: periodic type
In this case, we assume that Similarly, when we use Eq. (26) in Eq. (12), we obtain a system of algebraic equations from the coe cients of poly-nomial of ϕ(ξ ). By solving this algebraic system of equations, we get By solving the auxiliary equation ϕ (ξ ) = b − b ϕ (ξ ) and substituting together with (27) into Eq. (26), we get the solution of eq. (1) namely where ξ = x + y + z − b b α t α and < α ≤ . Fig. 4 depicts the 3D and 2D charts of the solution given by u (x, y, z, t) with the parameters a = . , b = − . , and b = .

. . Case 2: soliton type
Here, we assume that When we use Eq. (29) in Eq. (12), we obtain a system of algebraic equations from the coe cients of polynomial of ϕ(ξ ). By solving this algebraic system of equations, we get where By solving the auxiliary equation ϕ (ξ ) = b + b ϕ(ξ ) + b ϕ (ξ ) and substituting together with (30) into Eq. (29), we get the solution of Eq. (1) namely where ξ = x + y + z + b b α t α and < α ≤ . Fig. 5 depicts the 3D and 2D charts of the solution given by u (x, y, z, t) with the parameters a = . , b = − . , H = , and b = .

Conclusion
In this work, we have constructed exact traveling wave solutions for nonlinear conformable fractional evolution equations by using uni ed method. This method allows us to solve more nonlinear conformable fractional evolution equations in mathematical physics via the (3+1) dimensional conformable fractional Zakharov-Kuzetsov equation with power law nonlinearity. As a result, many new types of exact traveling wave solutions are obtained. These solutions include the solitary wave solutions, soliton wave solutions, elliptic wave solutions, and periodic (hyperbolic) wave rational solutions.