New Exact Solutions of the System of Equations for the Ion Sound and Langmuir Waves by ETEM

Abstract: This manuscript focuses attention on new exact solutions of the system of equations for the ion sound wave under the action of the ponderomotive force due to high-frequency field and for the Langmuir wave. The extended trial equation method (ETEM), which is one of the analytical methods, has been handled for finding exact solutions of the system of equations for the ion sound wave and the Langmuir wave. By using this method, exact solutions including the rational function solution, traveling wave solution, soliton solution, Jacobi elliptic function solution, hyperbolic function solution and periodic wave solution of this system of equations have been obtained. In addition, by using Mathematica Release 9, some graphical simulations were done to see the behavior of these solutions.


Introduction
The survey of new exact solutions for the ion sound wave and the Langmuir wave has a highly important position among the scientists.Many authors have worked on the Langmuir solitons.Degtyarev et al. have presented some properties of Langmuir solitons [1].Then, they have examined the Langmuir wave energy dissipation [2].Some authors have obtained the numerical simulations of Langmuir collapse [3][4][5][6].Benilov has demonstrated the stability of solitons by using the Zakharov equation which identifies the interaction between Langmuir and ion-sound waves [7].Zakharov et al. have exhibited the modeling of Langmuir turbulence [8].Dyachenko et al. have calculated computer simulations of Langmuir collapse [9].Rubenchik et al. have studied strong Langmuir turbulence in laser plasma [10].Musher et al. have submitted weak Langmuir turbulence [11].In addition, some authors have focused on Langmuir waves [12][13][14].Dodin et al. have tackled Langmuir wave evolution in nonstationary plasma [15].Zavlavsky et al. have considered spatial localization of Langmuir waves [16].In addition, Langmuir wave spectral energy densities have been derived from the electric field and compared to the weak turbulence results by Ratcliffe et al. [17].
We consider the system of equations for the ion sound wave under the action of the ponderomotive force due to high-frequency field and for the Langmuir wave [18]: where Ee ´iw p t is the normalized electric field of the Langmuir oscillation and n is the normalized density perturbation.The spatial variable x and the time variable t are also normalized appropriately [18].The system of Equation ( 1) for the ion sound and Langmuir waves has been submitted by Zakharov [19].Recently, this system has been investigated by some authors [20][21][22][23][24][25][26].
In this study, the basic interest is to constitute new exact solutions of the system of equations for the ion sound and Langmuir waves via extended trial equation method (ETEM).In Section 2, we mention basic facts of ETEM [27][28][29][30][31][32].In Section 3, we get new exact solutions of the system of equations for the ion sound and Langmuir waves via ETEM.

Basic Facts of the ETEM
Step 1.For a common nonlinear partial differential equation (NLPDE), perform the wave transformation where λ ‰ 0 and c ‰ 0. Replacing Equation (3) with Equation ( 2) reduces a nonlinear ordinary differential equation (NLODE), Step 2. Fulfill transformation and trial equation as the following: where Taking into consideration Equations ( 5) and ( 6), we can reach where φ pΓq and ψ pΓq are polynomials.Putting these terms into Equation ( 4) yields an equation of polynomial Ω pΓq of Γ: With regard to the balance principle, we can constitute a formula of θ, P and δ.We can gain some values of θ, P and δ.
Step 4. Reduce Equation ( 6) to basic integral form, Performing a complete discrimination system for polynomials to distinguish the roots of φ pΓq , we solve the infinite integral Equation (11) and classify the exact solutions of Equation (2) via Mathematica [33].

ETEM for the System of Equations for the Ion Sound and Langmuir Waves
In this section, we seek the exact solutions of the system of equations for the ion sound and Langmuir waves by using ETEM.
Remark 2. When the modulus l Ñ 0, then by using Equation ( 12), the Solution (36) can be reduced to the periodic wave solutions where α 2 " α 3 .
Remark 3. The exact solutions of Equation (1) were found via ETEM, and have been calculated by using Mathematica 9.As far as we know, the solutions of Equation ( 1) obtained in this study are new and are not observable in former literature.

Conclusions
In this paper, we obtain exact solutions of the system of equations for the ion sound and Langmuir waves by using ETEM.Then, for suitable parametric choices, we plot two and three dimensional graphics of some exact solutions of this system of equations by using Mathematica Release 9.This method supplies us to make complicated and tedious algebraic calculations.That is to say, the availability of computer programs such as Mathematica accelerates the tedious algebraic calculations.
The above results show that ETEM has been efficient for the analytical solutions of the system of equations for the ion sound and Langmuir waves.In addition, this method is a powerful mathematical tool in the way of finding new exact solutions.Thus, we can point out that ETEM has a key role in obtaining analytical solutions of NLPDEs.The graphical demonstrations such as Figures 1-6 clearly indicate effectiveness of the recommended method.We suggest that this method can also be applied to other NLPDEs.
α α = Remark 3. The exact solutions of Equation ( 1) were found via ETEM, and have been calculated by using Mathematica 9.As far as we know, the solutions of Equation ( 1) obtained in this study are new and are not observable in former literature.

Conclusions
In this paper, we obtain exact solutions of the system of equations for the ion sound and Langmuir waves by using ETEM.Then, for suitable parametric choices, we plot two and three dimensional graphics of some exact solutions of this system of equations by using Mathematica Release 9.This method supplies us to make complicated and tedious algebraic calculations.That is to say, the availability of computer programs such as Mathematica accelerates the tedious algebraic calculations.
The above results show that ETEM has been efficient for the analytical solutions of the system of equations for the ion sound and Langmuir waves.In addition, this method is a powerful mathematical tool in the way of finding new exact solutions.Thus, we can point out that ETEM has a key role in obtaining analytical solutions of NLPDEs.The graphical demonstrations such as Figures 1-6 clearly indicate effectiveness of the recommended method.We suggest that this method can also be applied to other NLPDEs.34) is denoted at m " τ 0 " 1,       34) is denoted at          36) is drawn at τ 0 " τ 1 " α 1 " 1, p " ´1, k " r " ξ 4 " α 2 " 2, α 3 " 3, ξ 3 " α 4 " 4, ´15 ă x ă 15, ´1 ă t ă 1, and the second graph illustrates n 5 px, tq in Equation (36) for ´15 ă x ă 15, t " 1.

Figure 2 . 3 ,
Figure 2. Graph of real values of

Figure 2 . 3 ,
Figure 2. Graph of real values of

3 ,
E x t in Equation (34) for