Symmetry reductions of the (3+1)$(3+1)$-dimensional modified Zakharov–Kuznetsov equation

This paper is concerned with the symmetry reductions of the (3 + 1)-dimensional modified Zakharov–Kuznetsov equation of ion-acoustic waves in a magnetized plasma. The direct symmetry method is applied to determine the symmetry and the corresponding vector field. Then, the considered equation is reduced to lower-dimensional equations with the aid of the obtained symmetry. At last, some exact solutions of the modified Zakharov–Kuznetsov equation are found in terms of the lower-dimensional equations.


Introduction
The Zakharov-Kuznetsov (ZK) equation [1] u t + uu x + 2 u x = 0 (1) was first proposed by Zakharov and Kuznetsov to describe the evolution of weakly nonlinear ion-acoustic waves in a plasma consisting of hot isothermal electrons and cold ions in the presence of a uniform magnetic field in the x direction. Equation (1) also appears in many other scientific fields including geochemistry, optical fiber, and solid state physics [2][3][4][5]. In [6], Shivamoggi provided a detailed discussion of the analytical properties of Eq. (1). Nawaz et al. [7] found appropriate solutions for the ZK equations with fully nonlinear dispersion by the homotopy analysis method. In 1999, Munro and Parkes considered a more realistic situation where the electrons are non-isothermal [8]. With an appropriately modified form of the electron number density given in [9], they showed that the reductive perturbation can lead to the following modified Zakharov-Kuznetsov (mZK) equation: 16(u tku x ) + 30u 1 2 u x + u xxx + u xyy + u xzz = 0, where k is a positive constant. Later, in [10] and [11], Munro and Parkes addressed the stability of solitary wave solutions and that of obliquely propagating solitary wave solutions to the mZK equation, respectively. In 2016, by using an extended direct algebraic method, Seadaway presented traveling wave solutions to the mZK equation and analyzed the stability for the electric fields and the electric field potentials [12]. It is noted that the mZK equation is a high dimensional nonlinear evolution equation and, thus, the study of its reduction problem is of theoretical interest. The Lie-group method, originally proposed by Sophus Lie, is a classical method to determine the symmetry reduction of partial differential equations (PDEs) [13][14][15][16]. During the past several decades, there have been many extensions of the Lie-group method such as the nonclassical Lie group method [17], the CK direct method [18], the direct symmetry method [19], and so on [20][21][22][23][24]. Among them, the direct symmetry method is an effective approach for seeking symmetry reductions. In [25] and [26], the method was used to investigate the Gardner-KP equation and the (2 + 1)-dimensional Jaulent-Miodek equation, respectively. To our knowledge, there is no result concerning the application of the direct symmetry method to the mZK equation partly due to its high dimension and nonlinear term u 1 2 u x , which motivates the present work.
Based on the above discussion, this paper considers the problem of seeking symmetry reductions of the mZK equation. In Sect. 2, with the help of the direct symmetry method, the symmetry and the corresponding vector field of the mZK equation are determined. In Sect. 3, by solving the symmetry equation, similarity transformations are constructed, which are applied to reduce the mZK equation to (2 + 1)-dimensional or even (1 + 1)dimensional equations. In Sect. 4, some exact solutions including trigonometric function solutions, hyperbolic function solutions, and Weierstrass function solutions of the mZK equation are presented in terms of the lower-dimensional equations. Finally, the conclusion is provided in Sect. 5.

Symmetry analysis
For an arbitrary nonlinear evolution equation where u x = ∂u ∂x . The function σ (x, t, u, u x , u t , . . .) is called a symmetry [27] of Eq. (3) if it satisfies the following equation for an arbitrary solution u(x, t): where Note that Eq. (4) is a linear PDE of the symmetry σ . Therefore the linear combination of symmetry σ is also a symmetry of Eq. (3). According to Eq. (4), the symmetry of mZK equation must satisfy Here, we set where a, b, c, d, e, and g are functions to be determined later. With the help of Maple, one can expand Eq. (5) by means of Eqs. (2) and (6). Then, taking the coefficients of u and those of the derivatives of u to zero yields the following twenty-one determining equation concerning a, b, c, d, e, and g: u xy : c zz + c yy + 2e y = 0, Solving the above equations yields where δ 0 , δ 1 , δ 2 , δ 3 , δ 4 , and δ 5 are arbitrary constants. Hence we obtain a general symmetry of the (3 + 1)-dimensional nonlinear mZK equation The corresponding vector field of the above symmetry can be expressed as which has the following infinitesimal generators: The commutation relation of these infinitesimal generators is given in Table 1.
In fact, it is not difficult to see that the symmetry of Eq. (13) satisfies where we set where a, b, c, e, and g are functions to be found later. To ensure that the expansion of Eq. (17) is true for an arbitrary solution f , we must take the coefficients of f and its deriva-tives to be zero. Hence we have According to the same procedure, it leads to Hence, Eq. (17) is reduced to From Eqs. (19)- (20), we obtain where λ 1 , λ 2 , and λ 3 are arbitrary constants. Thus we get a symmetry of Eq. (13) as follows: The corresponding characteristic equation of σ = 0 is From Eq. (23) we can obtain the following similarity variables φ, ϕ, and F: By using the obtained similarity variables, we obtain the further reduced equation of Eq. (13) 11ϕF ϕϕ + ϕ 2 + 1 F ϕϕϕ + 6φϕF φϕϕ + 16F φ + 25F ϕ + 30F ϕ √ F which is a (1+1)-dimensional nonlinear PDE. It is easy to find that the symmetry of Eq. (25) satisfies the following equation: Using the same direct symmetry method, we can reduce Eq. (25) to an ordinary differential equation (ODE). Here we omit them.
Remark 2 It is shown that (2 + 1)-dimensional Eq. (13) can be reduced to (1 + 1)dimensional partial differential equation Eq. (25). Similarly, Eqs. (14)- (16) can be discussed by the same method. In theory, Eq. (25) can be further reduced to an ODE. This problem will be discussed in our future work.