Separation Transformation and a Class of Exact Solutions to the Higher-Dimensional Klein-Gordon-Zakharov Equation

The separation transformation method is extended to the (n + 1)-dimensional Klein-Gordon-Zakharov equation describing the interaction of the Langmuir wave and the ion acoustic wave in plasma. We first reduce the (n + 1)-dimensional Klein-GordonZakharov equation to a set of partial differential equations and two nonlinear ordinary differential equations of the separation variables.Then the general solutions of the set of partial differential equations are given and the two nonlinear ordinary differential equations are solved by extendedF-expansionmethod. Finally, some new exact solutions of the (n + 1)-dimensional Klein-GordonZakharov equation are proposed explicitly by combining the separation transformation with the exact solutions of the separation variables. It is shown that, for the case of n ≥ 2, there is an arbitrary function in every exact solution, which may reveal more nontrivial nonlinear structures in the high-dimensional Klein-Gordon-Zakharov equation.


Introduction
The Klein-Gordon equation (sometimes called Klein-Gordon-Fock equation) [1] is a relativistic version of the Schrödinger equation.Its nonlinear counterpart is the nonlinear Klein-Gordon equation [2]: where  and  are constants, which has important applications in various fields.For example, it is attributed to the classical  4 field theory in the physics of elementary particles and fields, and it can describe the propagation of dislocations within crystals and the propagation of magnetic flux on a Josephson line, and so on.One extension of the nonlinear Klein-Gordon equation is the (1 + 1)-dimensional Klein-Gordon-Zakharov (KGZ) equation [3,4]: with (, ) as a complex function and V(, ) as a real function, which is a classical model describing the interaction of the Langmuir wave and the ion acoustic wave in plasma [3,4].The variable (, ) denotes the fast time scale component of electric field raised by electrons and the variable V(, ) denotes the deviation of ion density from its equilibrium.In recent years, some authors applied analytical and numerical methods [5][6][7][8] to solve the (1+1)-dimensional KGZ equation (2) and obtained many exact and numerical solutions.The high-dimensional extension of KGZ equation is important in real applications, so in this paper we would like to investigate the ( + 1)-dimensional KGZ equation: ) is the Laplacian operator and x ∈ R  .This equation is the generalization of the KGZ equation ( 2) and we will show that it has many exact solutions with an arbitrary function.
More recently, Wang [9] extended the separation transformation method proposed in [10][11][12] to the ( + 1)dimensional coupled nonlinear Klein-Gordon equations.Then Liu et al. [13] and we [14] further extended the separation transformation method to various high-dimensional nonlinear soliton equations and obtained explicitly many exact solutions with arbitrary functions.
In this paper, by means of the separation transformation approach [9][10][11][12][13][14] we derive the exact solutions of the ( + 1)-dimensional KGZ equation (3).The rest of this paper is organized as follows.In Section 2, a separation transformation is presented and the ( + 1)-dimensional KGZ equation ( 3) is reduced to a set of partial differential equations and two nonlinear ordinary differential equations.In Section 3, the two nonlinear ordinary differential equations are solved and some special exact solutions of the ( + 1)-dimensional KGZ equation ( 3) are constructed explicitly.Conclusions are presented in Section 4.

Separation Transformation and Its Application
The following proposition reveals the relationship between the exact solutions of the ( + 1)-dimensional KGZ equation ( 3) and two nonlinear ordinary differential equations (ODEs) along with a set of partial differential equations (PDEs).
We see that under the separation transformation ( 6) the ( + 1)-dimensional KGZ equation ( 3) is separated into two sets of differential equations, namely, the PDEs in (8) and ODEs in (9).If we can obtain the exact solutions of the differential equations ( 8) and ( 9), the explicitly exact solutions of the ( + 1)-dimensional KGZ equation ( 3) can be built immediately.In what follows, we solve the PDEs in (8) firstly.
It is remarked that when  ≥ 2, there is an arbitrary function (∑  =1     +  0  +  3 ) in each exact solution of the ( + 1)-dimensional KGZ equation (3), which may reveal abundant nonlinear structures in this nonlinear equation.

New Exact Solutions of the (𝑛+1)-
Dimensional KGZ Equation (3) In this section, we search for the exact solutions of the nonlinear ODEs in ( 9) by means of the -expansion method proposed by Wang et al. [15][16][17].Based on the explicit solutions of the ODEs in (11), many exact solutions of the ( + 1)-dimensional KGZ equation ( 3) are obtained explicitly via the separation transformation (6).
Integrating the first equation in (9) we have Thus the second equation in ( 9) becomes an ODE of () as In what follows, we solve the ODE ( 16) by using the extended -expansion method proposed by Wang et al. [15][16][17].In doing so, assume that the solution of ODE ( 16) is where  0 ,  1 , and  1 are constants to be determined and () satisfies the elliptic equation [18]: whose solutions in Jacobi elliptic function forms [18] are listed in Table 1 in the Appendix.Substituting (17) with (18) into ODE (16), we find that the variables  0 ,  1 ,  1 , and  have two groups of solutions.
Solution 7. Jacobi elliptic sn-ns-function solution is as follows: where the functions  and  are given by (11) for  = 1 and ( 12) with ( 13) for  ≥ 2.
Remark 2. It is noted that we can also list many other types of exact solutions for the ( + 1)-dimensional KGZ equation (3) by using the exact solutions of the elliptic equation ( 18) in Table 1.