New Exact Solutions of the System of Equations for

Mathematical and Computational Applications

Article

New Exact Solutions of the System of Equations for the Ion Sound and Langmuir Waves by ETEM Seyma Tuluce Demiray † and Hasan Bulut *,† Department of Mathematics, Firat University, Elazig 23119, Turkey; [email protected] * Correspondence: [email protected]; Tel.: +90-424-237-0000 † These authors contributed equally to this work. Academic Editor: Mehmet Pakdemirli Received: 19 February 2016; Accepted: 29 March 2016; Published: 2 April 2016

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. Keywords: the system of equations for the ion sound wave and the Langmuir wave; extended trial equation method; exact solutions; Mathematica Release 9

1. 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 [3–6]. 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 [12–14]. 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]: BE 1 B 2 E ` ´ nE “ 0, Bt 2 Bx2 2 2 B 2 |E|2 B n B n ´ ´ 2 “ 0, Bt2 Bx2 Bx2 i

(1)

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

Math. Comput. Appl. 2016, 21, 11; doi:10.3390/mca21020011

www.mdpi.com/journal/mca

Math. Comput. Appl. 2016, 21, 11

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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 [20–26]. 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 [27–32]. In Section 3, we get new exact solutions of the system of equations for the ion sound and Langmuir waves via ETEM. 2. Basic Facts of the ETEM Step 1. For a common nonlinear partial differential equation (NLPDE), P pu, ut , u x , u xx , . . .q “ 0

(2)

perform the wave transformation ¨ u px1 , x2 , . . . , x N , tq “ u pηq , η “ λ ˝

N ÿ

˛ x j ´ ct‚,

(3)

j“1

where λ ‰ 0 and c ‰ 0. Replacing Equation (3) with Equation (2) reduces a nonlinear ordinary differential equation (NLODE), ` ˘ N u, u1 , u2 , . . . “ 0. (4) Step 2. Fulfill transformation and trial equation as the following: u“

δ ÿ

τi Γi ,

(5)

i “0

where

` 1 ˘2 φ pΓq ξ Γθ ` . . . ` ξ 1 Γ ` ξ 0 Γ “ Λ pΓq “ “ θ P . ψ pΓq ζP Γ ` . . . ` ζ1 Γ ` ζ0

(6)

Taking into consideration Equations (5) and (6), we can reach ˜ ` 1 ˘2 φ pΓq u “ ψ pΓq φ1 pΓq ψ pΓq ´ φ pΓq ψ1 pΓq u “ 2ψ2 pΓq

˜

2

δ ÿ

δ ÿ

¸2 iτi Γ

i´1

,

(7)

i “0

˜

¸ iτi Γ

i ´1

i “0

φ pΓq ` ψ pΓq

δ ÿ

¸ i pi ´ 1q τi Γ

i ´2

,

(8)

i “0

where φ pΓq and ψ pΓq are polynomials. Putting these terms into Equation (4) yields an equation of polynomial Ω pΓq of Γ: Ω pΓq “ σs Γs ` . . . ` σ1 Γ ` σ0 “ 0. (9) 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 3. Letting the coefficients of Ω pΓq all be zero will construct an algebraic equations system: σi “ 0, i “ 0, . . . s.

(10)

Solving this equation system (10), we will find the values of ξ 0 , . . . , ξ θ ; ζ 0 , . . . , ζ P and τ0 , . . . , τδ . Step 4. Reduce Equation (6) to basic integral form, ż ˘ pη ´ η0 q “

dΓ a “ Λ pΓq

ż d

ψ pΓq dΓ. φ pΓq

(11)


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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]. 3. 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. In an effort to find traveling wave solutions of the Equation (1), we get the transformation by use of the wave variables E px, tq “ eiθ u pξq , n px, tq “ v pξq , θ “ kx ` mt, ξ “ px ` rt,

(12)

where k, m, p and r are arbitrary constants. Substituting Equations (13)–(15) into Equation (1), iEt “ ´meiθ u ` ireiθ u1 ,

(13)

Exx “ ´k2 eiθ u ` 2ipkeiθ u1 ` p2 eiθ u2 , ´ ¯ ´ ¯2 ntt “ r2 v2 , n xx “ p2 v2 , |E|2 “ p2 u2 .

(14)

xx

(15)

We obtain the following system: i pr ` pkq u1 pξq “ 0,

(16)

´ ¯ p2 u2 ´ 2m ` k2 u ´ 2uv “ 0, ´ ¯2 ´ ¯ r2 ´ p2 v2 ´ 2p2 u2 “ 0.

(17) (18)

By setting the integration constant to zero, we integrate function v with respect to ξ, we find v pξq “

2p2 u2 pξq . r 2 ´ p2

(19)

Putting Equation (19) into Equation (17) and by using Equation (16), we gain ´ ¯ ´ ¯´ ¯ p2 k2 ´ 1 u2 ´ k2 ´ 1 2m ` k2 u ´ 4u3 “ 0,

(20)

where the prime remarks the derivative with respect to ξ. Substituting Equations (5) and (8) into Equation (20) and using the balance principle, the formula is found as θ “ 2δ` P `2. (21) In order to gain exact solutions of Equation (1), if we take P“ 0, δ “ 1 and θ “ 4 in Equation (21), then ` ˘ ` ˘ ` 1 ˘2 τ12 ξ 0 ` Γξ 1 ` Γ2 ξ 2 ` Γ3 ξ 3 ` Γ4 ξ 4 2 τ1 ξ 1 ` 2Γξ 2 ` 3Γ2 ξ 3 ` 4Γ3 ξ 4 v “ ,v “ , ζ0 2ζ 0

(22)

where ξ 4 ‰ 0, ζ 0 ‰ 0. Solving the algebraic equation system (10) provides ` ˘ p2 k2 ´ 1 ξ 32 ξ 33 ´ 4ξ 2 ξ 3 ξ 4 ξ0 “ ξ0, ξ2 “ ξ2, ξ3 “ ξ3, ξ4 “ ξ4, ξ1 “ ´ , ζ0 “ , 2 2 `8ξ 4 2 ˘ 2 32ξ 4 τ0 4ξ τ0 k2 2 ´3ξ 3 ` 8ξ 2 ξ 4 τ0 ` ˘ τ0 “ τ0 , τ1 “ 4 , m “ ´ ` . ξ3 2 k2 ´ 1 ξ 32

(23)


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Setting these results into Equations (6) and (11), we have ż ˘ pη ´ η0 q “ A

c

dΓ , ξ0 ξ1 ξ2 2 ξ3 3 4 ` Γ` Γ ` Γ `Γ ξ4 ξ4 ξ4 ξ4

(24)

d

` ˘ p2 k2 ´ 1 ξ 32 where A “ . 32ξ 4 τ02 Integrating Equation (24), we find the solutions of Equation (1) as the following: ˘ pη ´ η0 q “ ´

A , Γ ´ α1

(25)

d

Γ ´ α2 , α2 ą α1 , (26) Γ ´ α1 ˇ ˇ ˇ Γ ´ α1 ˇ A ˇ , α ą α2 , (27) ˘ pη ´ η0 q “ ln ˇˇ α1 ´ α2 Γ ´ α2 ˇ 1 ˇa ˇ ˇ pΓ ´ α q pα ´ α q ´ apΓ ´ α q pα ´ α q ˇ 2A ˇ 2 3 3 2 ˇ 1 1 a ˘ pη ´ η0 q “ a ln ˇ a ˇ , α1 ą α2 ą α3 , (28) pα1 ´ α2 q pα1 ´ α3 q ˇ pΓ ´ α2 q pα1 ´ α3 q ` pΓ ´ α3 q pα1 ´ α2 q ˇ 2A ˘ pη ´ η0 q “ α1 ´ α2

2A ˘ pη ´ η0 q “ a F pϕ, lq , α1 ą α2 ą α3 ą α4 , pα1 ´ α3 q pα2 ´ α4 q

(29)

where żϕ F pϕ, lq “ 0

d dψ

b , ϕ “ arcsin 1 ´ l 2 sin2 ψ

pΓ ´ α1 q pα2 ´ α4 q 2 pα2 ´ α3 q pα1 ´ α4 q ,l “ . pΓ ´ α2 q pα1 ´ α4 q pα1 ´ α3 q pα2 ´ α4 q

(30)

In addition, α1 , α2 , α3 and α4 are the roots of the polynomial equation Γ4 `

ξ3 3 ξ2 2 ξ1 ξ0 Γ ` Γ ` Γ` “ 0. ξ4 ξ4 ξ4 ξ4

(31)

Substituting the solutions Equations (25)–(29) into Equation (5) and by using Equation (12), the solutions of Equation (1) are obtained rational function solutions, ˆ ˙ A1 E1 px, tq “ ˘ , px ` rt ˙ ˆ ˙ ˆ 2 2p2 A1 n1 px, tq “ ˘ , px ` rt r 2 ´ p2 eipkx`mtq

˜ E2 px, tq “ ˆ n2 px, tq “

eipkx`mtq 2p2 2 r ´ p2

4A2 pα2 ´ α1 q τ1

(32)

¸

, 4A2 ´ rpα1 ´ α2 q ppx ` rtqs2 ¸2 ˙˜ 4A2 pα2 ´ α1 q τ1 4A2 ´ rpα1 ´ α2 q ppx ` rtqs2

(33) ,

traveling wave solutions ˆ " „ *˙ pα2 ´ α1 q τ1 pα1 ´ α2 q ppx ` rtq E3 px, tq “ eipkx`mtq 1 ˘ coth , 2 A ˆ ˙ ˆ " *˙ „ 2 2p2 pα2 ´ α1 q τ1 pα1 ´ α2 q ppx ` rtq n3 px, tq “ 1 ˘ coth , 2 A r 2 ´ p2

(34)


soliton solutions

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A2 , E4 px, tq “ eipkx`mtq pD ` cosh rB ppx ` rtqsq ˆ ˙ ˆ ˙ 2 2p2 A2 n4 px, tq “ , D ` cosh rB ppx ` rtqs r 2 ´ p2

(35)

and Jacobi elliptic function solutions A3 ˘, E5 px, tq “ eipkx`mtq ` M ` Nsn2 pϕ, lq ˙ˆ ˙2 ˆ A3 2p2 n5 px, tq “ , r 2 ´ p2 M ` Nsn2 pϕ, lq ˆ where A1 “ τ1 A, A2 “ A3

(36)

a pα1 ´ α2 q pα1 ´ α3 q 2α1 ´ α2 ´ α3 ,B “ ,D “ , A α3 ´ α2 pα2 ´ α3 q pα1 ´ α4 q α4 ´ α2 , N “ α1 ´ α4 , l 2 “ , pα1 ´ α3 q pα2 ´ α4 q

2τ1 pα1 ´ α2 q pα1 ´ α3 q α3 ´ α2

˙

p2τ1 pα1 ´ α3 q pα4 ´ α2 qq , M “ a ˘ pα1 ´ α3 q pα2 ´ α4 q ppx ` rtq . Here, A2 is the amplitude of the soliton, and B is the inverse ϕ “ 2A width of the solitons. Thus, the solitons exist for τ1 ă 0. “

Remark 1. When the modulus l Ñ 1, then by using Equation (12), the Solution (36) can be converted into the hyperbolic function solutions A3 «a ff¸ , pα q pα q ´ α ´ α 3 2 1 4 ppx ` rtq M ` Ntanh2 2A ˛2 ¨

E6 px, tq “ eipkx`mtq ˜

ˆ n6 px, tq “

2p2

˙˚ ˚ ˚ 2 2 r ´p ˚ ˝

(37)

‹ ‹ A3 «a ff ‹ ‹ , pα1 ´ α3 q pα2 ´ α4 q ‚ 2 ppx ` rtq M ` Ntanh 2A

where α3 “ α4 . Remark 2. When the modulus l Ñ 0, then by using Equation (12), the Solution (36) can be reduced to the periodic wave solutions A3

E7 px, tq “ eipkx`mtq ˜

«a M ` N sin2

¨ ˆ n7 px, tq “

2p2

˙˚ ˚ ˚ 2 2 r ´p ˚ ˝

ff¸ , pα1 ´ α3 q pα2 ´ α4 q ppx ` rtq 2A ˛2

A3 «a pα1 ´ α3 q pα2 ´ α4 q ppx ` rtq M ` N sin2 2A

(38)

‹ ‹ ff ‹ ‹ , ‚

where α2 “ α3 . 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.

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 former Math. Comput.inAppl. 2016,literature. 21, 11 6 of 9 4. Conclusions 4. Conclusions In this paper, we obtain exact solutions of the system of equations for the ion sound and In this paper, solutions system parametric of equationschoices, for the ion and Langmuir Langmuir waveswe byobtain usingexact ETEM. Then, of forthesuitable wesound plot two and three waves by using ETEM.of Then, suitable parametric plot two and three dimensional dimensional graphics somefor exact solutions of thischoices, system we of equations by using Mathematica graphics ofThis somemethod exact solutions system of equations using Mathematica Release 9.That Thisis Release 9. supplies of us this to make complicated andbytedious algebraic calculations. method supplies us to of make complicated andsuch tedious algebraic calculations. That is to algebraic say, the to say, the availability computer programs as Mathematica accelerates the tedious availability of computer programs such as Mathematica accelerates the tedious algebraic calculations. calculations. The been efficient forfor thethe analytical solutions of the system of Theabove aboveresults resultsshow showthat thatETEM ETEMhas has been efficient analytical solutions of the system equations for the sound Langmuir waves. Inwaves. addition, methodthis is a powerful mathematical of equations forion the ion and sound and Langmuir Inthis addition, method is a powerful tool in the waytool of finding newof exact solutions. Thus, we can point ETEMout hasthat a key rolehas in mathematical in the way finding new exact solutions. Thus, out we that can point ETEM obtaining solutions of NLPDEs. graphical demonstrations such as Figures clearly a key roleanalytical in obtaining analytical solutions The of NLPDEs. The graphical demonstrations such1–6 as Figures indicate effectiveness of the recommended method. Wemethod. suggest We thatsuggest this method can method also be applied 1–6 clearly indicate effectiveness of the recommended that this can also to NLPDEs. beother applied to other NLPDEs.

6 4 2

35

30

25

20

15

10

5 2 4 6

Figure 1. 1.Graph Graph of imaginary Equation (34) at is mindicated at Figure of imaginary valuesvalues of E3 px,oftq inE3Equation is indicated “ τ0 “ 1, ( x, t ) in (34) τ1m= “τ 0=k 1,“τ 1= ξ 3k=“ ξξ3=4 “ “ p3, p “ 4,