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

Authors: Mehmet Ekici; Durgun Duran; Abdullah Sonmezoglu
Journal: ISRN Computational Mathematics
 (2013)

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## Abstract

We introduce a new version of the trial equation method for solving nonintegrable partial differential equations in mathematical physics. Some exact solutions including soliton solutions and rational and elliptic function solutions to the Klein-Gordon-Zakharov equation with power law nonlinearity in (1 + 2) dimensions are obtained by this method.


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## Body

## 1. Introduction

In recent years there have been many works on the qualitative research of the global solutions for the Klein-Gordon-Zakharov (KGZ) equations [1–4]. 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 sine-cosine 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 equation method [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.


## 2. 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 1.
For a given nonlinear partial differential equation with rank inhomogeneous(1)P(u,ut,ux,uxx,…)=0,
take the wave transformation
(2)u(x1,…,xN,t)=u(η),η=γ(∑j=1N‍xj-ct),
where γ≠0 and c≠0. Substituting (2) into (1) yields a nonlinear ordinary differential equation
(3)N(u,u′,u′′,…)=0.Step 2.
Take transformation and trial equation as follows:(4)u=∑i=0δ‍τiΓi,
in which
(5)(Γ′)2=Λ(Γ)=Φ(Γ)Ψ(Γ)=ξθΓθ+⋯+ξ1Γ+ξ0ζϵΓϵ+⋯+ζ1Γ+ζ0,
where τi(i=0,…,δ), ξi(i=0,…,θ), and ζi(i=0,…,ϵ) are constants. Using the relations (4) and (5), we can find
(6)(u′)2=Φ(Γ)Ψ(Γ)(∑i=0δ‍iτiΓi-1)2,u′′=Φ′(Γ)Ψ(Γ)-Φ(Γ)Ψ′(Γ)2Ψ2(Γ)(∑i=0δ‍iτiΓi-1)+Φ(Γ)Ψ(Γ)(∑i=0δ‍i(i-1)τiΓi-2),
where Φ(Γ) and Ψ(Γ) are polynomials. Substituting these terms into (3) yields an equation of polynomial Ω(Γ) of Γ as follows:
(7)Ω(Γ)=ϱsΓs+⋯+ϱ1Γ+ϱ0=0.
According to the balance principle we can determine a relation of θ, ϵ, and δ. We can take some values of θ, ϵ, and δ.Step 3.
Let the coefficients ofΩ(Γ) all be zero, this will yield the following algebraic equations system:
(8)ϱi=0,i=0,…,s.
Solving this equation system (8), we will determine the values of ξ0,…,ξθ; ζ0,…,ζϵ, and τ0,…,τδ.Step 4.
Reduce (5) to the elementary integral form as follows:
(9)±(η-η0)=∫dΓΛ(Γ)=∫Ψ(Γ)Φ(Γ)dΓ.
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.


## 3. 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.


### 3.1. 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]
(10)qtt-λ2(qxx+qyy)+q+rq+α|q|2mq=0,(11)rtt-λ2(rxx+ryy)=(|q|2m)xx+(|q|2m)yy.
Here, the dependent variables are q and r, while the independent variables are x, y, and t 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 m>0 dictates the power law nonlinearity, while λ and α are constants. Here, in (10) and (11), q is a complex valued function while r 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 ion-acoustic waves in plasmas [32, 33]. For solving (10) and (11) with the trial equation method, using the wave variables
(12)q(x,y,t)=u(η)eiϕ,r(x,y,t)=v(η)ϕ=-κ1x-κ2y+ωt+ς,η=β1x+β2y-ct,
where κ1, κ2, ω, ς, β1, β2, and c are real constants, (10) and (11) are converted to the system of ODEs
(13)(wc-λ2(κ1β1+κ2β2))u′(η)=0,(14)(1-ω2+λ2(κ12+κ22))u(η)+(c2-λ2(β12+β22))u′′(η)+αu2m+1(η)+u(η)v(η)=0,(15)(c2-λ2(β12+β22))v′′(η)-(β12+β22)(u2m)′′(η)=0,
where primes denote the derivatives with respect to η. Equation (15) is then integrated term by term two times where integration constants are considered zero. This converts it into
(16)v(η)=β12+β22c2-λ2(β12+β22)u2m(η).
Substituting (16) into (14) gives
(17)(1-ω2+λ2(κ12+κ22))u(η)+(c2-λ2(β12+β22))u′′(η)+(α+β12+β22c2-λ2(β12+β22))u2m+1(η)=0.
Equation (17), with the transformation
(18)u(η)=V1/m(η),
reduces to
(19)P(c2-Q)VV′′+(P2-P)(c2-Q)(V′)2+(R-w2)V2+(α+Qλ2(c2-Q))V4=0,
where
(20)P=1m,Q=λ2(β12+β22),R=1+λ2(κ12+κ22).
Substituting (6) into (19) and using balance principle yields θ=ϵ+2δ+2. If we take θ=4, ϵ=0, and δ=1, then
(21)(V′)2=τ12(ξ4Γ4+ξ3Γ3+ξ2Γ2+ξ1Γ+ξ0)ζ0,
where ξ4≠0 and ζ0≠0. Solving the algebraic equation system (8) yields
(22)ξ0=-ζ0τ04PQ+λ2ζ0τ02(Q-c2)(ατ02P+(1+P)(R-ω2))λ2τ12P2(1+P)(c2-Q)2,ξ3=-4ζ0τ0τ1(Q+αλ2(c2-Q))λ2P(1+P)(c2-Q)2,ξ1=-4ζ0τ03PQ+2λ2ζ0τ0(Q-c2)(2ατ02P+(1+P)(R-ω2))λ2τ1P2(1+P)(c2-Q)2,ξ4=-ζ0τ12(Q+αλ2(c2-Q))λ2P(1+P)(c2-Q)2,ξ2=-6ζ0τ02PQ+λ2ζ0(Q-c2)(6ατ02P+(1+P)(R-ω2))λ2P2(1+P)(c2-Q)2,ζ0=ζ0,τ0=τ0,τ1=τ1.
Also from (13), it can be seen that wc=λ2(κ1β1+κ2β2). Substituting these results into (5) and (9), we can write
(23)±(η-η0)=-λ2P(1+P)(c2-Q)2τ12(Q+αλ2(c2-Q))×∫dΓΓ4+ℓ3Γ3+ℓ2Γ2+ℓ1Γ+ℓ0,
where
(24)ℓ3=4τ0τ1,ℓ2=-6τ02PQ+λ2(Q-c2)(6ατ02P+(1+P)(R-ω2))τ12P(-Q+αλ2(Q-c2)),ℓ1=-4τ03PQ+2λ2τ0(Q-c2)(2ατ02P+(1+P)(R-ω2))τ13P(-Q+αλ2(Q-c2)),ℓ0=ζ0τ04PQ-λ2ζ0τ02(Q-c2)(ατ02P+(1+P)(R-ω2))ζ0τ14P(Q+αλ2(c2-Q)).Integrating (23), we obtain the solutions to (10) and (11) as follows:
(25)±(η-η0)=-BΓ-α1,(26)±(η-η0)=2Bα1-α2Γ-α2Γ-α1,α2>α1,(27)±(η-η0)=Bα1-α2ln|Γ-α1Γ-α2|,(28)±(η-η0)=B(α1-α2)(α1-α3)×ln|(Γ-α2)(α1-α3)-(Γ-α3)(α1-α2)(Γ-α2)(α1-α3)+(Γ-α3)(α1-α2)|,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiα1>α2>α3,(29)±(η-η0)=2B(α1-α3)(α2-α4)F(φ,l),iiiiiiiiiiiiiiiiiα1>α2>α3>α4,
where
(30)B=-λ2P(1+P)(c2-Q)2τ12(Q+αλ2(c2-Q)),F(φ,l)=∫0φ‍dψ1-l2sin2ψ,φ=arcsin(Γ-α1)(α2-α4)(Γ-α2)(α1-α4),l2=(α2-α3)(α1-α4)(α1-α3)(α2-α4).
Also α1, α2, α3, and α4 are the roots of the polynomial equation
(31)Γ4+ξ3ξ4Γ3+ξ2ξ4Γ2+ξ1ξ4Γ+ξ0ξ4=0.
Substituting the solutions (25)–(28) into (4) and (18), we obtain, respectively,
(32)q(x,y,t)={τ0+τ1α1±τ1Bβ1x+β2y-ct-η0}1/meiϕ,(33)r(x,y,t)=β12+β22c2-λ2(β12+β22)×{τ0+τ1α1±τ1Bβ1x+β2y-ct-η0}2,(34)q(x,y,t)={B2[(β1x+β2y-ct-η0)]2τ0+τ1α1+4B2(α2-α1)τ14B2-[(α1-α2)(β1x+β2y-ct-η0)]2}1/meiϕ,(35)r(x,y,t)=β12+β22c2-λ2(β12+β22)×{B2[(β1x+β2y-ct-η0)]2τ0+τ1α1+4B2(α2-α1)τ14B2-[(α1-α2)(β1x+β2y-ct-η0)]2}2,(36)q(x,y,t)={((α1-α2B)-1)-1τ0+τ1α2+((α2-α1)τ1)=-×(exp(α1-α2B(β1x+β2y-ct-η0))-1)-1}1/meiϕ,(37)r(x,y,t)=β12+β22c2-λ2(β12+β22)×{×(exp(α1-α2B(β1x+β2y-ct-η0))-1)-1α1-α22Bτ0+τ1α2+((α2-α1)τ1)×(exp(α1-α2B(β1x+β2y-ct-η0))-1)-1}2,(38)q(x,y,t)={×(exp(α1-α2B(β1x+β2y-ct-η0))-1)-1τ0+τ1α1+((α1-α2)τ1)×(exp(α1-α2B(β1x+β2y-ct-η0))-1)-1}1/meiϕ,(39)r(x,y,t)=β12+β22c2-λ2(β12+β22)×{×(exp(α1-α2B(β1x+β2y-ct-η0))-1)-1α1-α22Bτ0+τ1α1+((α1-α2)τ1)×(exp(α1-α2B(β1x+β2y-ct-η0))-1)-1}2,(40)q(x,y,t)={(α1-α2)(α1-α3)Bτ0+τ1α1-(2(α1-α2)(α1-α3)τ1)×((α1-α2)(α1-α3)B2α1-α2-α3+(α3-α2)×cosh((α1-α2)(α1-α3)B(α1-α2)(α1-α3)B×(β1x+β2y-ct)(α1-α2)(α1-α3)2B))-1}1/meiϕ,(41)r(x,y,t)=β12+β22c2-λ2(β12+β22)×{×(β1x+β2y-ct))(α1-α2)(α1-α3)B)-1τ0+τ1α1-(2(α1-α2)(α1-α3)τ1)×(×(β1x+β2y-ct))(α1-α2)(α1-α3)B)-12α1-α2-α3+(α3-α2)×cosh(×(β1x+β2y-ct))(α1-α2)(α1-α3)B)-1(α1-α2)(α1-α3)B×(β1x+β2y-ct))(α1-α2)(α1-α3)B)-1}2.
If we take τ0=-τ1α1 and η0=0, then the solutions (32)–(41) can reduce to rational function solutions
(42)q(x,y,t)=(±τ1Bβ1x+β2y-ct)1/mei(-κ1x-κ2y+ωt+ς),r(x,y,t)=Υ(τ1Bβ1x+β2y-ct)2,q(x,y,t)={4B2(α2-α1)τ14B2-[(α1-α2)(β1x+β2y-ct)]2}1/m×ei(-κ1x-κ2y+ωt+ς),r(x,y,t)=Υ{4B2(α2-α1)τ14B2-[(α1-α2)(β1x+β2y-ct)]2}2,
traveling wave solutions
(43)q(x,y,t)={(α2-α1)τ12{1∓coth[α1-α22B(β1x+β2y-ct)]}}1/m×ei(-κ1x-κ2y+ωt+ς),r(x,y,t)=Υ{(α2-α1)τ12{1∓coth[α1-α22B(β1x+β2y-ct)]}}2,
and soliton solutions
(44)q(x,y,t)=A1(D+cosh[B1(β1x+β2y-ct)])1/m×ei(-κ1x-κ2y+ωt+ς),(45)r(x,y,t)=ΥA2(D+cosh[B1(β1x+β2y-ct)])2,
where
(46)Υ=β12+β22c2-λ2(β12+β22),A1=(2(α1-α2)(α1-α3)τ1α2-α3)1/m,A2=A12m,B1=(α1-α2)(α1-α3)B,D=2α1-α2-α3α3-α2.
Here, A1 and A2 are the amplitudes of the solitons, β1 and β2 are the inverse widths of the solitons in the x- and y-directions, respectively, while c is the velocity of the soliton. Also, κ1 and κ2 are the soliton frequencies in the x- and y-directions, respectively, and ω is the soliton wave number, while ς is the phase constant. Thus, we can say that the solitons exist for τ1>0.Remark.  Ismail and Biswas obtained 1-soliton solution of KGZ equation in [31], we obtained soliton solution of this equation. In case 2α1=α2+α3, our soliton solutions in (44) and (45) reduce to 1-soliton solutions in [31].In Figures1, 2, and 3, we give profiles of numerical soliton solutions of (44) and (45) for various values of parameters.Figure 1
Profile of a numerical solution of (44) at m=2, β1=β2=1, A1=2, B1=0.8, and D<0, while ct=1.Figure 2
Profile of a numerical solution of (45) at β1=β2=1, A2=4, B1=0.5, D<0, and ct=1, while Υ>0.Figure 3
Profile of a numerical solution of (45) at β1=β2=1, A2=4, B1=0.5, D<0, and ct=1, while Υ<0.


## 3.1. 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]
(10)qtt-λ2(qxx+qyy)+q+rq+α|q|2mq=0,(11)rtt-λ2(rxx+ryy)=(|q|2m)xx+(|q|2m)yy.
Here, the dependent variables are q and r, while the independent variables are x, y, and t 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 m>0 dictates the power law nonlinearity, while λ and α are constants. Here, in (10) and (11), q is a complex valued function while r 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 ion-acoustic waves in plasmas [32, 33]. For solving (10) and (11) with the trial equation method, using the wave variables
(12)q(x,y,t)=u(η)eiϕ,r(x,y,t)=v(η)ϕ=-κ1x-κ2y+ωt+ς,η=β1x+β2y-ct,
where κ1, κ2, ω, ς, β1, β2, and c are real constants, (10) and (11) are converted to the system of ODEs
(13)(wc-λ2(κ1β1+κ2β2))u′(η)=0,(14)(1-ω2+λ2(κ12+κ22))u(η)+(c2-λ2(β12+β22))u′′(η)+αu2m+1(η)+u(η)v(η)=0,(15)(c2-λ2(β12+β22))v′′(η)-(β12+β22)(u2m)′′(η)=0,
where primes denote the derivatives with respect to η. Equation (15) is then integrated term by term two times where integration constants are considered zero. This converts it into
(16)v(η)=β12+β22c2-λ2(β12+β22)u2m(η).
Substituting (16) into (14) gives
(17)(1-ω2+λ2(κ12+κ22))u(η)+(c2-λ2(β12+β22))u′′(η)+(α+β12+β22c2-λ2(β12+β22))u2m+1(η)=0.
Equation (17), with the transformation
(18)u(η)=V1/m(η),
reduces to
(19)P(c2-Q)VV′′+(P2-P)(c2-Q)(V′)2+(R-w2)V2+(α+Qλ2(c2-Q))V4=0,
where
(20)P=1m,Q=λ2(β12+β22),R=1+λ2(κ12+κ22).
Substituting (6) into (19) and using balance principle yields θ=ϵ+2δ+2. If we take θ=4, ϵ=0, and δ=1, then
(21)(V′)2=τ12(ξ4Γ4+ξ3Γ3+ξ2Γ2+ξ1Γ+ξ0)ζ0,
where ξ4≠0 and ζ0≠0. Solving the algebraic equation system (8) yields
(22)ξ0=-ζ0τ04PQ+λ2ζ0τ02(Q-c2)(ατ02P+(1+P)(R-ω2))λ2τ12P2(1+P)(c2-Q)2,ξ3=-4ζ0τ0τ1(Q+αλ2(c2-Q))λ2P(1+P)(c2-Q)2,ξ1=-4ζ0τ03PQ+2λ2ζ0τ0(Q-c2)(2ατ02P+(1+P)(R-ω2))λ2τ1P2(1+P)(c2-Q)2,ξ4=-ζ0τ12(Q+αλ2(c2-Q))λ2P(1+P)(c2-Q)2,ξ2=-6ζ0τ02PQ+λ2ζ0(Q-c2)(6ατ02P+(1+P)(R-ω2))λ2P2(1+P)(c2-Q)2,ζ0=ζ0,τ0=τ0,τ1=τ1.
Also from (13), it can be seen that wc=λ2(κ1β1+κ2β2). Substituting these results into (5) and (9), we can write
(23)±(η-η0)=-λ2P(1+P)(c2-Q)2τ12(Q+αλ2(c2-Q))×∫dΓΓ4+ℓ3Γ3+ℓ2Γ2+ℓ1Γ+ℓ0,
where
(24)ℓ3=4τ0τ1,ℓ2=-6τ02PQ+λ2(Q-c2)(6ατ02P+(1+P)(R-ω2))τ12P(-Q+αλ2(Q-c2)),ℓ1=-4τ03PQ+2λ2τ0(Q-c2)(2ατ02P+(1+P)(R-ω2))τ13P(-Q+αλ2(Q-c2)),ℓ0=ζ0τ04PQ-λ2ζ0τ02(Q-c2)(ατ02P+(1+P)(R-ω2))ζ0τ14P(Q+αλ2(c2-Q)).Integrating (23), we obtain the solutions to (10) and (11) as follows:
(25)±(η-η0)=-BΓ-α1,(26)±(η-η0)=2Bα1-α2Γ-α2Γ-α1,α2>α1,(27)±(η-η0)=Bα1-α2ln|Γ-α1Γ-α2|,(28)±(η-η0)=B(α1-α2)(α1-α3)×ln|(Γ-α2)(α1-α3)-(Γ-α3)(α1-α2)(Γ-α2)(α1-α3)+(Γ-α3)(α1-α2)|,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiα1>α2>α3,(29)±(η-η0)=2B(α1-α3)(α2-α4)F(φ,l),iiiiiiiiiiiiiiiiiα1>α2>α3>α4,
where
(30)B=-λ2P(1+P)(c2-Q)2τ12(Q+αλ2(c2-Q)),F(φ,l)=∫0φ‍dψ1-l2sin2ψ,φ=arcsin(Γ-α1)(α2-α4)(Γ-α2)(α1-α4),l2=(α2-α3)(α1-α4)(α1-α3)(α2-α4).
Also α1, α2, α3, and α4 are the roots of the polynomial equation
(31)Γ4+ξ3ξ4Γ3+ξ2ξ4Γ2+ξ1ξ4Γ+ξ0ξ4=0.
Substituting the solutions (25)–(28) into (4) and (18), we obtain, respectively,
(32)q(x,y,t)={τ0+τ1α1±τ1Bβ1x+β2y-ct-η0}1/meiϕ,(33)r(x,y,t)=β12+β22c2-λ2(β12+β22)×{τ0+τ1α1±τ1Bβ1x+β2y-ct-η0}2,(34)q(x,y,t)={B2[(β1x+β2y-ct-η0)]2τ0+τ1α1+4B2(α2-α1)τ14B2-[(α1-α2)(β1x+β2y-ct-η0)]2}1/meiϕ,(35)r(x,y,t)=β12+β22c2-λ2(β12+β22)×{B2[(β1x+β2y-ct-η0)]2τ0+τ1α1+4B2(α2-α1)τ14B2-[(α1-α2)(β1x+β2y-ct-η0)]2}2,(36)q(x,y,t)={((α1-α2B)-1)-1τ0+τ1α2+((α2-α1)τ1)=-×(exp(α1-α2B(β1x+β2y-ct-η0))-1)-1}1/meiϕ,(37)r(x,y,t)=β12+β22c2-λ2(β12+β22)×{×(exp(α1-α2B(β1x+β2y-ct-η0))-1)-1α1-α22Bτ0+τ1α2+((α2-α1)τ1)×(exp(α1-α2B(β1x+β2y-ct-η0))-1)-1}2,(38)q(x,y,t)={×(exp(α1-α2B(β1x+β2y-ct-η0))-1)-1τ0+τ1α1+((α1-α2)τ1)×(exp(α1-α2B(β1x+β2y-ct-η0))-1)-1}1/meiϕ,(39)r(x,y,t)=β12+β22c2-λ2(β12+β22)×{×(exp(α1-α2B(β1x+β2y-ct-η0))-1)-1α1-α22Bτ0+τ1α1+((α1-α2)τ1)×(exp(α1-α2B(β1x+β2y-ct-η0))-1)-1}2,(40)q(x,y,t)={(α1-α2)(α1-α3)Bτ0+τ1α1-(2(α1-α2)(α1-α3)τ1)×((α1-α2)(α1-α3)B2α1-α2-α3+(α3-α2)×cosh((α1-α2)(α1-α3)B(α1-α2)(α1-α3)B×(β1x+β2y-ct)(α1-α2)(α1-α3)2B))-1}1/meiϕ,(41)r(x,y,t)=β12+β22c2-λ2(β12+β22)×{×(β1x+β2y-ct))(α1-α2)(α1-α3)B)-1τ0+τ1α1-(2(α1-α2)(α1-α3)τ1)×(×(β1x+β2y-ct))(α1-α2)(α1-α3)B)-12α1-α2-α3+(α3-α2)×cosh(×(β1x+β2y-ct))(α1-α2)(α1-α3)B)-1(α1-α2)(α1-α3)B×(β1x+β2y-ct))(α1-α2)(α1-α3)B)-1}2.
If we take τ0=-τ1α1 and η0=0, then the solutions (32)–(41) can reduce to rational function solutions
(42)q(x,y,t)=(±τ1Bβ1x+β2y-ct)1/mei(-κ1x-κ2y+ωt+ς),r(x,y,t)=Υ(τ1Bβ1x+β2y-ct)2,q(x,y,t)={4B2(α2-α1)τ14B2-[(α1-α2)(β1x+β2y-ct)]2}1/m×ei(-κ1x-κ2y+ωt+ς),r(x,y,t)=Υ{4B2(α2-α1)τ14B2-[(α1-α2)(β1x+β2y-ct)]2}2,
traveling wave solutions
(43)q(x,y,t)={(α2-α1)τ12{1∓coth[α1-α22B(β1x+β2y-ct)]}}1/m×ei(-κ1x-κ2y+ωt+ς),r(x,y,t)=Υ{(α2-α1)τ12{1∓coth[α1-α22B(β1x+β2y-ct)]}}2,
and soliton solutions
(44)q(x,y,t)=A1(D+cosh[B1(β1x+β2y-ct)])1/m×ei(-κ1x-κ2y+ωt+ς),(45)r(x,y,t)=ΥA2(D+cosh[B1(β1x+β2y-ct)])2,
where
(46)Υ=β12+β22c2-λ2(β12+β22),A1=(2(α1-α2)(α1-α3)τ1α2-α3)1/m,A2=A12m,B1=(α1-α2)(α1-α3)B,D=2α1-α2-α3α3-α2.
Here, A1 and A2 are the amplitudes of the solitons, β1 and β2 are the inverse widths of the solitons in the x- and y-directions, respectively, while c is the velocity of the soliton. Also, κ1 and κ2 are the soliton frequencies in the x- and y-directions, respectively, and ω is the soliton wave number, while ς is the phase constant. Thus, we can say that the solitons exist for τ1>0.Remark.  Ismail and Biswas obtained 1-soliton solution of KGZ equation in [31], we obtained soliton solution of this equation. In case 2α1=α2+α3, our soliton solutions in (44) and (45) reduce to 1-soliton solutions in [31].In Figures1, 2, and 3, we give profiles of numerical soliton solutions of (44) and (45) for various values of parameters.Figure 1
Profile of a numerical solution of (44) at m=2, β1=β2=1, A1=2, B1=0.8, and D<0, while ct=1.Figure 2
Profile of a numerical solution of (45) at β1=β2=1, A2=4, B1=0.5, D<0, and ct=1, while Υ>0.Figure 3
Profile of a numerical solution of (45) at β1=β2=1, A2=4, B1=0.5, D<0, and ct=1, while Υ<0.


## 4. 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 different value of arbitrary parameters. So we can choose appropriate parameter value to obtain solutions which we need in applications. The method can also be employed to solve a large number of other nonlinear evolution equations, such as nonlinear reaction-diffusion equation, the long-short wave resonance equation, the shallow water wave equation, Whitham-Broer-Kaup equation, variant Boussinesq equation, double Sine-Gardon equation, and Dodd-Bullough-Mikhailov equation.



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*Source: 716279-2013-12-03.xml*