# Extended Jacobi Elliptic Function Expansion Method to the ZK-MEW Equation

Authors: Weimin Zhang
Journal: International Journal of Differential Equations
 (2011)

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## Abstract

The extended Jacobi elliptic function expansion method is applied for Zakharov-Kuznetsov-modified equal-width (ZK-MEW) equation. With the aid of symbolic computation, we construct some new Jacobi elliptic doubly periodic wave solutions and the corresponding solitary wave solutions and triangular functional (singly periodic) solutions.


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## Body

## 1. Introduction

It is one of the most important tasks to seek the exact solutions of nonlinear equation in the study of the nonlinear equations. Up to now, many powerful methods have been developed such as inverse scattering transformation [1], Backlund transformation [2], Hirota bilinear method [3], homogeneous balance method [4], extended tanh-function method [5], Jacobi elliptic function expansion method [6] and Ma’s transformed rational function method [7].Recently [8], an extended-tanh method is used to establish exact travelling wave solution of the Zakharov-Kuznetsov-modified equal-width (ZK-MEW) equation. In this paper, an extended Jacobi elliptic function expansion method is employed to construct some new exact solutions of the Zakharov-Kuznetsov-modified equal-width (ZK-MEW) equation.As known, the Zakharov-Kuznetsov (ZK) equation are given by(1.1)(ut+auux+uxxx)x+uyy=0,(1.2)ut+auux+(∇2u)x=0,
where ∇2=∂x2+∂y2+∂z2 is the isotropic Laplacian. The ZK equation governs the behavior of weakly nonlinear ion-acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field [9]. In [9], the ZK equation is solved by the sine-cosine and the tanh-function methods. In [10], the numbers of solitary waves, periodic waves, and kink waves of the modified Zakharov-Kuznetsov equation are obtained.The regularized long wave (RLW) equation given by(1.3)ut+ux+12(u2)x-uxxt=0,-∞<x<+∞,t>0,
appears in many physical applications and has been studied in [11]. Gardner et al. [12] solved the equal width equation by a Petrov Galerkin method using quadratic B-spline spatial finite-elements.The modified equal width (MEW) equation given by(1.4)ut+3u2ux-βuxxt=0,
has been discussed in [11]. The MEW equation is related to the RLW equation. This equation has solitary waves with both positive and negative amplitudes. The two-dimensional ZK-MEW equation which first appeared in [13] is given by (1.5)ut+a(u3)x+(buxt+ruyy)x=0,
where u=u(x,y,t), a,b,r are constants. In [13], some exact solutions of the ZK-MEW equation (1.5) was obtained by using the tanh and sine-cosine methods. More detailed description for ZK-MEW equation (1.5) the reader can find in paper [13]. In this paper, we will give some new solutions of Jacobi elliptic function type of ZK-MEW equation by using an extended Jacobi elliptic function method.The remainder of the paper is organized as follows. In Section2, we briefly describe the extended Jacobi elliptic function expansion method. In Section 3, we apply this method to ZK-MEW equation to construct exact solutions. Finally, some conclusions are given in Section 4.


## 2. The Extended Jacobi Elliptic Function Expansion Method

In this section, the extended Jacobi elliptic function expansion method is proposed in [14]. Consider a given nonlinear wave equation, say in two variables(2.1)F(u,ux,ut,uxx,uxt,utt,…)=0.
We make the transformation(2.2)u=u(x,t)=U(ξ),ξ=k(x-ct),
where k is a constant to be determined later. Then (2.1) reduceds to a nonlinear ordinary differential equation (ODE) under(2.2)(2.3)G(U,Uξ,Uξξ,Uξξξ,…)=0.
By the extended Jacobi elliptic function expansion method, introduce the following ansatz     (2.4)U(ξ)=∑j=-MNajYj(ξ),
where M,N,aj(j=-M,…,N) are constants to be determined later, Y is an Jacobi elliptic function, namely, Y=Y(ξ)=snξ=sn(ξ,m)orcn(ξ,m)ordn(ξ,m), m(0<m<1) is the modulus of Jacobian elliptic functions. Positive integer M,N can be determined by balancing the highest-order linear term with the nonlinear term in (2.3). After this, substituting (2.4) into (2.3), we can obtain a system of algebraic equations for aj(j=-M,…,N). Solving the above-mentioned equations with the Mathematica Software, then aj(j=-M,…,N) can be determined. Substituting these obtained results into (2.4), then a general form of Jacobi elliptic function solution of (2.1) can be given.


## 3. ZK-MEW Equation

In this section, we employ the extended Jacobi elliptic function expansion method to ZK-MEW equation that is given by (1.5). The transformation u=U(ξ),ξ=λ(x+μy-σt) converts (1.5) into ODE(3.1)-σUξ+a(U3)ξ+(rλ2μ2-bλ2σ)Uξξξ=0.
Integrating (3.1) and setting the constant of integration to zero, we obtain(3.2)-σU+a(U3)+(rλ2μ2-bλ2σ)Uξξ=0.
Substituting (2.4) into (3.2) to balance U3 with Uξξ, we find M=N=1. Thus, the solution admits in the form(3.3)U(ξ)=a-1Y-1(ξ)+a0+a1Y(ξ),
where a-1,a0,a1 are constants to be determined later.Notice that(3.4)d(snξ)dξ=sn′ξ=cnξdnξ,d(cnξ)dξ=cn′ξ=-snξdnξ,d(dnξ)dξ=dn′ξ=-msnξcnξ,cn2ξ=1-sn2ξ,dn2ξ=1-msn2ξ.


### 3.1. The Case ofY=Y(ξ)=snξ=sn(ξ,m)

SubstitutingY=Y(ξ)=snξ=sn(ξ,m) and (3.3) into (3.2), making use of (3.4), we obtain a system of algebraic equations, for a-1,a0,a1, and  λ of the following form:(3.5)2λ2(rμ2-bσ)a-1+aa-13=0,3aa-12a0=0,-a-1[(1+m)rλ2μ2-(-σ+b(1+m)λ2)σ-2aa02]+3aa-12a1=0,-σa0+aa03+6aa-1a0a1=0,(-1-m)rλ2μ2a1+[-1+b(1+m)λ2]σa1+3aa02a1+3aa-1a12=0.
Solving the system of the algebraic equations with the aid of Mathematica we can distinguish two cases, namely the following.Case 1.
(3.6)a1=0,a0=0,a-1=2σa(1+m),λ=σ(1+m)(bσ-rμ2).Case 2.
(3.7)a1=σ(1+m)a,a-1=σa(1+m),a0=0,λ=σ2(1+m)(bσ-rμ2).Substituting (3.6), (3.7) into (3.3), respectively, yield, the following solutions of ZK-MEW equation:(3.8)u(x,y,t)=2σa(1+m)sn-1[σ(1+m)(bσ-rμ2)(x+μy-σt),m],(3.9)u(x,y,t)=σ(1+m)asn[σ2(1+m)(bσ-rμ2)(x+μy-σt),m]+σa(1+m)sn-1[σ2(1+m)(bσ-rμ2)(x+μy-σt),m].
Notice that m→1, snξ→tanhξ, and m→0, snξ→sinξ, we can obtain solitary wave solutions and sin-wave solutions from (3.8) and (3.9), respectively,(3.10a)u(x,y,t)=σacoth⁡[σ2(bσ-rμ2)(x+μy-σt)],(3.10b)u(x,y,t)=2σasin-1[σbσ-rμ2(x+μy-σt)],(3.11a)u(x,y,t)=2σatanh[σ2(bσ-rμ2)(x+μy-σt)]+σ2acoth⁡[σ2(bσ-rμ2)(x+μy-σt)],(3.11b)u(x,y,t)=σasin[σ2(bσ-rμ2)(x+μy-σt)]+σasin-1[σ2(bσ-rμ2)(x+μy-σt)].Here we only give the graph of (3.8) (see Figure 1) and (3.10a) (see Figure 2) and the other graphs of equations are similar to discussing.(a) The graph of (3.8) with a=r=σ=1,b=2,m=0.5 at t=1, and (b) is its plane when y=0.
(a)(b)(a) The graph of (3.10a) with a=r=σ=1,b=2 at t=1 and (b) is its plane when y=0 (the case is the same to (3.8) when m=1).
(a)(b)


### 3.2. The Case ofY=Y(ξ)=dnξ=dn(ξ,m)

The analysis proceeds of this case is as for Section3.1. Substituting Y=Y(ξ)=dnξ=dn(ξ,m) and (3.3) into (3.2), making use of (3.4), we obtain a system of algebraic equations, for a-1,a0,a1 and λ of the following form:(3.12)2(-1+m)λ2(rμ2-bσ)a-1+aa-13=0,3aa-12a0=0,-a-1[(-2+m)rλ2μ2+σ+2bλ2σ-bmλ2σ-3aa02]+3aa-12a1=0,-σa0+aa03+6aa-1a0a1=0,(2-m)rλ2μ2a1+[-1+b(-2+m)λ2]σa1+3aa02a1+3aa-1a12=0,a1[2λ2(-rμ2+bσ)+aa12]=0.
Solving the system of the algebraic equations with the aid of Mathematica we can distinguish three cases, namely, The following.Case 1.
(3.13)a-1=0,a0=0,a1=2σa(2-m),λ=σ(2-m)(rμ2-bσ).Case 2.
(3.14)a-1=2(1-m)σa(2-61-m-m),a0=0,a1=2σa(2-61-m-m),λ=σ(2-61-m-m)(bσ-rμ2).Case 3.
(3.15)a-1=2(1-m)σa(2-m),a0=0,a1=0,λ=σ(2-m)(rμ2-bσ).Substituting (3.13), (3.14), and (3.15) into (3.3), respectively, yields the following solutions of ZK-MEW equation:(3.16)u(x,y,t)=2σa(2-m)dn[σ(2-m)(rμ2-bσ)(x+μy-σt),m],(3.17)u(x,y,t)=2(1-m)σa(2-61-m-m)dn-1[σ(2-61-m-m)(bσ-rμ2)(x+μy-σt),m]+2σa(2-61-m-m)dn[σ(2-61-m-m)(bσ-rμ2)(x+μy-σt),m],(3.18)u(x,y,t)=2(1-m)σa(2-m)dn-1[σ(2-m)(rμ2-bσ)(x+μy-σt),m].
Notice that m→1, dnξ→sechξ, thus we can obtain solitary wave of solutions of ZK-MEW equation from (3.16) and (3.17), respectively,(3.19)u(x,y,t)=2σasech[σ±(rμ2-bσ)(x+μy-σt)].
Here we only give the graph of (3.17) (see Figure 3) and (3.19) (see Figure 4) and the other graphs of equations are similar to discussing.(a) The graph of (3.17) with a=r=σ=1,b=2,m=0.99 at t=1, and (b) is its plane when y=0.
(a)(b)(a) The solitary wave graph of (3.19) with a=r=σ=1,b=2 at t=1, and (b) is its plane when y=0 (the case is the same to (3.17) when m=1).
(a)(b)


### 3.3. The Case ofY=Y(ξ)=cnξ=cn(ξ,m)

The analysis proceeds of this case is as for Sections 3.1 and 3.2. SubstitutingY=Y(ξ)=cnξ=cn(ξ,m) and (3.3) into (3.2), making use of (3.4), we obtain a system of algebraic equations, for a-1,a0,a1, and λ of the following form:(3.20)-2(-1+m)λ2(rμ2-bσ)a-1+aa-13=0,3aa-12a0=0,a-1[(-1+2m)rλ2μ2-σ+bσ(1-2m)λ2+3aa02]+3aa-12a1=0,-σa0+aa03+6aa-1a0a1=0,(-1+2m)rλ2μ2a1-σa1+bσ(1-2m)λ2a1+3aa02a1+3aa12a-1=0,3aa12a0=0,a1[2mλ2(-rμ2+bσ)+aa12]=0.
Solving the system of the algebraic equations (3.20) with the aid of Mathematica, we can distinguish three cases, namely, the following.Case 1.
(3.21)a0=0,a1=2mσa(-1+2m),a-1=0,λ=σ(-1+2m)(rμ2-bσ).Case 2.
(3.22)a0=0,a-1=2(1-m)σ-a(-1+2m),a1=0,λ=σ(-1+2m)(rμ2-bσ).Case 3.
(3.23)a0=σa,a1=-2mσa(1-2m),a-1=0,λ=2σ(1-2m)(rμ2-bσ).Substituting (3.21), (3.22), and (3.23) into (3.3), respectively, yield the following solutions of ZK-MEW equation:(3.24)u(x,y,t)=2mσa(-1+2m)cn[σ(-1+2m)(rμ2-bσ)(x+μy-σt),m],(3.25)u(x,y,t)=2(1-m)σ-a(-1+2m)cn-1[σ(-1+2m)(rμ2-bσ)(x+μy-σt),m],(3.26)u(x,y,t)=σa-2mσa(1-2m)cn[2σ(1-2m)(rμ2-bσ)(x+μy-σt),m].
Especially, when m→1, cnξ→sechξ and m→0, cnξ→cosξ,  thus we can obtain solutions of ZK-MEW equation from (3.24) and(3.25)(3.27)u(x,y,t)=2σasech[σrμ2-bσ(x+μy-σt)],(3.28)u(x,y,t)=2σacos⁡-1⁡[σ-rμ2+bσ(x+μy-σt)].Remark 3.1.
In these solutions, (3.10a), (3.19) and (3.27) have been obtain in [8], the others solutions are new solutions for the ZK-MEW equation.


## 3.1. The Case ofY=Y(ξ)=snξ=sn(ξ,m)

SubstitutingY=Y(ξ)=snξ=sn(ξ,m) and (3.3) into (3.2), making use of (3.4), we obtain a system of algebraic equations, for a-1,a0,a1, and  λ of the following form:(3.5)2λ2(rμ2-bσ)a-1+aa-13=0,3aa-12a0=0,-a-1[(1+m)rλ2μ2-(-σ+b(1+m)λ2)σ-2aa02]+3aa-12a1=0,-σa0+aa03+6aa-1a0a1=0,(-1-m)rλ2μ2a1+[-1+b(1+m)λ2]σa1+3aa02a1+3aa-1a12=0.
Solving the system of the algebraic equations with the aid of Mathematica we can distinguish two cases, namely the following.Case 1.
(3.6)a1=0,a0=0,a-1=2σa(1+m),λ=σ(1+m)(bσ-rμ2).Case 2.
(3.7)a1=σ(1+m)a,a-1=σa(1+m),a0=0,λ=σ2(1+m)(bσ-rμ2).Substituting (3.6), (3.7) into (3.3), respectively, yield, the following solutions of ZK-MEW equation:(3.8)u(x,y,t)=2σa(1+m)sn-1[σ(1+m)(bσ-rμ2)(x+μy-σt),m],(3.9)u(x,y,t)=σ(1+m)asn[σ2(1+m)(bσ-rμ2)(x+μy-σt),m]+σa(1+m)sn-1[σ2(1+m)(bσ-rμ2)(x+μy-σt),m].
Notice that m→1, snξ→tanhξ, and m→0, snξ→sinξ, we can obtain solitary wave solutions and sin-wave solutions from (3.8) and (3.9), respectively,(3.10a)u(x,y,t)=σacoth⁡[σ2(bσ-rμ2)(x+μy-σt)],(3.10b)u(x,y,t)=2σasin-1[σbσ-rμ2(x+μy-σt)],(3.11a)u(x,y,t)=2σatanh[σ2(bσ-rμ2)(x+μy-σt)]+σ2acoth⁡[σ2(bσ-rμ2)(x+μy-σt)],(3.11b)u(x,y,t)=σasin[σ2(bσ-rμ2)(x+μy-σt)]+σasin-1[σ2(bσ-rμ2)(x+μy-σt)].Here we only give the graph of (3.8) (see Figure 1) and (3.10a) (see Figure 2) and the other graphs of equations are similar to discussing.(a) The graph of (3.8) with a=r=σ=1,b=2,m=0.5 at t=1, and (b) is its plane when y=0.
(a)(b)(a) The graph of (3.10a) with a=r=σ=1,b=2 at t=1 and (b) is its plane when y=0 (the case is the same to (3.8) when m=1).
(a)(b)


## 3.2. The Case ofY=Y(ξ)=dnξ=dn(ξ,m)

The analysis proceeds of this case is as for Section3.1. Substituting Y=Y(ξ)=dnξ=dn(ξ,m) and (3.3) into (3.2), making use of (3.4), we obtain a system of algebraic equations, for a-1,a0,a1 and λ of the following form:(3.12)2(-1+m)λ2(rμ2-bσ)a-1+aa-13=0,3aa-12a0=0,-a-1[(-2+m)rλ2μ2+σ+2bλ2σ-bmλ2σ-3aa02]+3aa-12a1=0,-σa0+aa03+6aa-1a0a1=0,(2-m)rλ2μ2a1+[-1+b(-2+m)λ2]σa1+3aa02a1+3aa-1a12=0,a1[2λ2(-rμ2+bσ)+aa12]=0.
Solving the system of the algebraic equations with the aid of Mathematica we can distinguish three cases, namely, The following.Case 1.
(3.13)a-1=0,a0=0,a1=2σa(2-m),λ=σ(2-m)(rμ2-bσ).Case 2.
(3.14)a-1=2(1-m)σa(2-61-m-m),a0=0,a1=2σa(2-61-m-m),λ=σ(2-61-m-m)(bσ-rμ2).Case 3.
(3.15)a-1=2(1-m)σa(2-m),a0=0,a1=0,λ=σ(2-m)(rμ2-bσ).Substituting (3.13), (3.14), and (3.15) into (3.3), respectively, yields the following solutions of ZK-MEW equation:(3.16)u(x,y,t)=2σa(2-m)dn[σ(2-m)(rμ2-bσ)(x+μy-σt),m],(3.17)u(x,y,t)=2(1-m)σa(2-61-m-m)dn-1[σ(2-61-m-m)(bσ-rμ2)(x+μy-σt),m]+2σa(2-61-m-m)dn[σ(2-61-m-m)(bσ-rμ2)(x+μy-σt),m],(3.18)u(x,y,t)=2(1-m)σa(2-m)dn-1[σ(2-m)(rμ2-bσ)(x+μy-σt),m].
Notice that m→1, dnξ→sechξ, thus we can obtain solitary wave of solutions of ZK-MEW equation from (3.16) and (3.17), respectively,(3.19)u(x,y,t)=2σasech[σ±(rμ2-bσ)(x+μy-σt)].
Here we only give the graph of (3.17) (see Figure 3) and (3.19) (see Figure 4) and the other graphs of equations are similar to discussing.(a) The graph of (3.17) with a=r=σ=1,b=2,m=0.99 at t=1, and (b) is its plane when y=0.
(a)(b)(a) The solitary wave graph of (3.19) with a=r=σ=1,b=2 at t=1, and (b) is its plane when y=0 (the case is the same to (3.17) when m=1).
(a)(b)


## 3.3. The Case ofY=Y(ξ)=cnξ=cn(ξ,m)

The analysis proceeds of this case is as for Sections 3.1 and 3.2. SubstitutingY=Y(ξ)=cnξ=cn(ξ,m) and (3.3) into (3.2), making use of (3.4), we obtain a system of algebraic equations, for a-1,a0,a1, and λ of the following form:(3.20)-2(-1+m)λ2(rμ2-bσ)a-1+aa-13=0,3aa-12a0=0,a-1[(-1+2m)rλ2μ2-σ+bσ(1-2m)λ2+3aa02]+3aa-12a1=0,-σa0+aa03+6aa-1a0a1=0,(-1+2m)rλ2μ2a1-σa1+bσ(1-2m)λ2a1+3aa02a1+3aa12a-1=0,3aa12a0=0,a1[2mλ2(-rμ2+bσ)+aa12]=0.
Solving the system of the algebraic equations (3.20) with the aid of Mathematica, we can distinguish three cases, namely, the following.Case 1.
(3.21)a0=0,a1=2mσa(-1+2m),a-1=0,λ=σ(-1+2m)(rμ2-bσ).Case 2.
(3.22)a0=0,a-1=2(1-m)σ-a(-1+2m),a1=0,λ=σ(-1+2m)(rμ2-bσ).Case 3.
(3.23)a0=σa,a1=-2mσa(1-2m),a-1=0,λ=2σ(1-2m)(rμ2-bσ).Substituting (3.21), (3.22), and (3.23) into (3.3), respectively, yield the following solutions of ZK-MEW equation:(3.24)u(x,y,t)=2mσa(-1+2m)cn[σ(-1+2m)(rμ2-bσ)(x+μy-σt),m],(3.25)u(x,y,t)=2(1-m)σ-a(-1+2m)cn-1[σ(-1+2m)(rμ2-bσ)(x+μy-σt),m],(3.26)u(x,y,t)=σa-2mσa(1-2m)cn[2σ(1-2m)(rμ2-bσ)(x+μy-σt),m].
Especially, when m→1, cnξ→sechξ and m→0, cnξ→cosξ,  thus we can obtain solutions of ZK-MEW equation from (3.24) and(3.25)(3.27)u(x,y,t)=2σasech[σrμ2-bσ(x+μy-σt)],(3.28)u(x,y,t)=2σacos⁡-1⁡[σ-rμ2+bσ(x+μy-σt)].Remark 3.1.
In these solutions, (3.10a), (3.19) and (3.27) have been obtain in [8], the others solutions are new solutions for the ZK-MEW equation.


## 4. Conclusions

The extended Jacobi elliptic function expansion method was directly and effectively employed to find travelling wave solutions of the nonlinear ZK-MEW equation. Using the method, we found some new solutions of Jacobi elliptic function type that were not obtained by the sine-cosine method, the extended tanh-method, the mapping method, and other methods. In the limiting case of the Jacobi elliptic function (namely, modulus setting 0 or 1), we also obtained the solutions of sin-type, cos-tye, tanh-type, and sech-type. The extended Jacobi elliptic function expansion method can be applied to some other nonlinear equation and gives more solutions.The ZK-MEW equation was first appeared in Wazwaz’s paper [13] in 2005. To my acknowledge, its many properties, such as integrability, Lax pairs, and multisoliton solutions, have not been studied. The study of these properties is a very signification work and is our task research in the future.



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