Soliton Solutions of the Klein-Gordon-Zakharov Equation with Power Law Nonlinearity

In recent years there have been many works on the qualitative research of the global solutions for the Klein-GordonZakharov (KGZ) equations [1–4]. Chen considered orbital stability of solitary waves for the KGZ equations in [5]. More recently, some exact solutions for the Zakharov equations are obtained by using different methods [6–9]. These solutions are not general and by no means exhaust all possibilities. They are only some particular solutions within some specific parameters choices. The aim of this paper is to find the new and more general explicit and exact special solutions of the KGZ equations. We obtain various of explicit and exact special solutions of the KGZ equations by using the extended trial equation method. These solutions include that of the solitary wave solutions of the singular traveling wave solutions and solitary wave solutions of rational function type. Solving nonlinear evolution equations has become a valuable task in many scientific areas including applied mathematics as well as the physical sciences and engineering. Many powerful methods, such as the Backlund transformation, the inverse scattering method [10], bilinear transformation, the tanh-sech method [11], the extended tanh method, the pseudospectral method [12], the trial function and the sinecosine method [13], Hirota method [14], tanh-coth method [15, 16], the exponential function method [17], (G/G)expansion method [18, 19], homogeneous balance method [20], and the trial equationmethod [21–30] have been used to investigate nonlinear partial differential equations problems. There are a lot of nonlinear evolution equations that are integrated using these and other mathematical methods. In this paper, KGZ equations will be studied by extended trial equation. By virtue of the solitary wave ansatz method, an exact soliton solution will be obtained. The extended trial equation method will be employed to back up our analysis in obtaining exact solutions with distinct physical structures.


Introduction
In recent years there have been many works on the qualitative research of the global solutions for the Klein-Gordon-Zakharov (KGZ) equations [1][2][3][4].Chen considered orbital stability of solitary waves for the KGZ equations in [5].More recently, some exact solutions for the Zakharov equations are obtained by using different methods [6][7][8][9].These solutions are not general and by no means exhaust all possibilities.They are only some particular solutions within some specific parameters choices.
The aim of this paper is to find the new and more general explicit and exact special solutions of the KGZ equations.We obtain various of explicit and exact special solutions of the KGZ equations by using the extended trial equation method.These solutions include that of the solitary wave solutions of the singular traveling wave solutions and solitary wave solutions of rational function type.
In this paper, KGZ equations will be studied by extended trial equation.By virtue of the solitary wave ansatz method, an exact soliton solution will be obtained.The extended trial equation method will be employed to back up our analysis in obtaining exact solutions with distinct physical structures.

The Extended Trial Equation Method
The main steps of an extended trial equation method for the nonlinear partial differential equations with higher order nonlinearity are outlined as follows.
Step 4. Reduce (5) to the elementary integral form as follows: Using a complete discrimination system for polynomial to classify the roots of Φ(Γ), we solve the infinite integral ( 9) and obtain the exact solutions to (3).Furthermore, we can write the exact traveling wave solutions to (1), respectively.

Mathematical Analysis
We introduce the KGZ equation with power law nonlinearity in (1+2) dimensions and its soliton solution by extended trial equation method and show its numerical solution at a fixed point.

The KGZ Equation in (1 + 2) Dimensions.
The dimensionless form of the KGZ equation in (1 + 2) dimensions that will be studied in this subsection is given by [31] Here, the dependent variables are  and , while the independent variables are , , and  which are, respectively, referred to as the spatial variables and temporal variable.
Power law nonlinearity arises in nonlinear plasmas that solves the problem of small K-condensation in weak turbulence theory.It also arises in the context of nonlinear optics.The parameter  > 0 dictates the power law nonlinearity, while  and  are constants.Here, in (10) and (11),  is a complex valued function while  is a real valued function.Equations ( 10) and ( 11) together appear in the area of Plasma Physics.They describe the interaction of Langmuir waves and ionacoustic waves in plasmas [32,33].For solving (10) and (11) with the trial equation method, using the wave variables where  1 ,  2 , , ,  1 ,  2 , and  are real constants, ( 10) and ( 11) are converted to the system of ODEs where primes denote the derivatives with respect to .Equation ( 15) is then integrated term by term two times where integration constants are considered zero.This converts it into Substituting ( 16) into (14) gives Equation ( 17), with the transformation reduces to where Substituting ( 6) into ( 19) and using balance principle yields  =  + 2 + 2. If we take  = 4,  = 0, and  = 1, then where  4 ̸ = 0 and  0 ̸ = 0. Solving the algebraic equation system (8) yields Also from (13), it can be seen that  =  2 ( 1  1 +  2  2 ).Substituting these results into (5) and ( 9), we can write where Integrating (23), we obtain the solutions to ( 10) and ( 11) as follows: where
In Figures 1, 2, and 3, we give profiles of numerical soliton solutions of (44) and (45) for various values of parameters.

Conclusion
We adopt the extended trial equation method to obtain soliton solutions of the KGZ equations in plasma physics.We obtain some more general solitary wave solutions of the KGZ equations.It not only produces the same solutions but also can pick up what we believe to be new solutions missed by other authors.The results indicate the KGZ equations admit soliton solutions with some arbitrary parameters.The type of exact solitary wave solution is different along with