New Periodic Solutions for a Class of Zakharov Equations

Through applying the Jacobian elliptic function method, we obtain the periodic solution for a series of nonlinear Zakharov equations, which contain Klein-Gordon Zakharov equations, Zakharov equations, and Zakharov-Rubenchik equations.


Introduction
For most of nonlinear evolution equations, we have many methods to obtain their exactly solutions, such as hyperbolic function expansion method [1], the transformation method [2], the trial function method [3], the automated method [4], and the extended tanh-function method [5].But these methods can only obtain solitary wave solution and cannot be used to deduce periodic solutions.The Jacobian function method provides a way to find periodic solutions for some nonlinear evolution equations.In particular, in the research of plasma physics theory, quantum mechanics, fluid mechanics, and optical fiber communication, we frequently meet kinds of Zakharov equations.
Hence, in this paper, inspired by Angulo Pava's work [6], we are concerned to obtain exact periodic solutions of a series of Zakharov equations, and Zakharov-Rubenchik equations, (2)

The Periodic Solution for Klein-Gordon Zakharov Equations
The Klein-Gordon Zakharov equations are used to show the interaction between langmuir wave and ion wave in the plasma, which has the following form: where (, ) denotes the biggest moment scale component produced by electron in electric field.(, ) denotes the speed of deviations between the ions at any position and that at equilibrium position.Now, we suppose that it possesses solitary wave solutions of the following form: where  is a traveling wave speed and  2 < 1 is a constant.
In what follows, we are concerned with the periodic solution of (3); thus we need to require  1 = 0.
Here,  = √ 2/2(1 −  2 ).So By returning to initial variable, we obtain that is a dnoidal solution of (10).Furthermore, dn has fundamental period 2(); that is, dn( + 2; ) = dn(; ), and () is the complete elliptic integral of first kind.So the dnoidal wave solution  has fundamental period, , given by So, by applying the method of the Jacobian elliptic function and inspired by Angulo Pava's ideas, we obtain that (3) has the periodic traveling solution of the following form: Moreover,  and  2 can be also rewritten as the following form: Next we will show that, for an arbitrary but fixed , √ 1 − ( We can obtain that there Advances in Mathematical Physics 3 is a unique  2,0 =  2,0 ( 0 ) ∈ (0, √1 −  2 0 − ) such that ( 2 ) = ( 2 ()) =  is a fundamental period for the dnoidal wave solution (18).Theorem 1.Let  > 0 be arbitrary but fixed.
Proof.Based on the ideas establish in Angulo Pava's work [6], we will give a brief proof.Now, we consider the open set We define Φ : Ω →  by Here, In what follows, we will prove that Φ/ 2 < 0.
From (24), we can obtain that From  2 , we can deduce ( 2 , ) is a strictly decreasing function of  2 .

Conclusion
Inspired by Angulo Pava's idea, by applying Jacobian elliptic function method, we have obtained new period wave solution for Klein-Gordon Zakharov equations, Zakharov equations, and Zakharov-Rubenchik equations.In particular, the solutions of ( 18), (47), and (58) were not found in the previous work.The method can help to look for periodic solution for a class of nonlinear equations.