International Journal of Differential Equations

International Journal of Differential Equations / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 451420 | 11 pages | https://doi.org/10.1155/2011/451420

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

Academic Editor: Wen Xiu Ma
Received03 May 2011
Accepted25 Jun 2011
Published04 Sep 2011

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.

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 𝑢𝑡+ğ‘Žğ‘¢ğ‘¢ğ‘¥+𝑢𝑥𝑥𝑥𝑥+𝑢𝑦𝑦𝑢=0,(1.1)𝑡+ğ‘Žğ‘¢ğ‘¢ğ‘¥+∇2𝑢𝑥=0,(1.2) where ∇2=𝜕2𝑥+𝜕2𝑦+𝜕2𝑧 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 𝑢𝑡+𝑢𝑥+12𝑢2𝑥−𝑢𝑥𝑥𝑡=0,−∞<𝑥<+∞,𝑡>0,(1.3) 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 by𝑢𝑡+3𝑢2𝑢𝑥−𝛽𝑢𝑥𝑥𝑡=0,(1.4) 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 𝑢𝑡𝑢+ğ‘Ž3𝑥+𝑏𝑢𝑥𝑡+𝑟𝑢𝑦𝑦𝑥=0,(1.5) where 𝑢=𝑢(𝑥,𝑦,𝑡), ğ‘Ž,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 Section 2, 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 variables𝐹𝑢,𝑢𝑥,𝑢𝑡,𝑢𝑥𝑥,𝑢𝑥𝑡,𝑢𝑡𝑡,…=0.(2.1) We make the transformation𝑢=𝑢(𝑥,𝑡)=𝑈(𝜉),𝜉=𝑘(𝑥−𝑐𝑡),(2.2) where 𝑘 is a constant to be determined later. Then (2.1) reduceds to a nonlinear ordinary differential equation (ODE) under(2.2)𝐺𝑈,𝑈𝜉,𝑈𝜉𝜉,𝑈𝜉𝜉𝜉,…=0.(2.3) By the extended Jacobi elliptic function expansion method, introduce the following ansatz 𝑈(𝜉)=𝑁𝑗=âˆ’ğ‘€ğ‘Žğ‘—ğ‘Œğ‘—(𝜉),(2.4) where 𝑀,𝑁,ğ‘Žğ‘—(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 ğ‘Žğ‘—(𝑗=−𝑀,…,𝑁). Solving the above-mentioned equations with the Mathematica Software, then ğ‘Žğ‘—(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+ğœ‡ğ‘¦âˆ’ğœŽğ‘¡) converts (1.5) into ODEâˆ’ğœŽğ‘ˆğœ‰î€·ğ‘ˆ+ğ‘Ž3𝜉+𝑟𝜆2𝜇2−𝑏𝜆2ğœŽî€¸ğ‘ˆğœ‰ğœ‰ğœ‰=0.(3.1) Integrating (3.1) and setting the constant of integration to zero, we obtainî€·ğ‘ˆâˆ’ğœŽğ‘ˆ+ğ‘Ž3+𝑟𝜆2𝜇2−𝑏𝜆2ğœŽî€¸ğ‘ˆğœ‰ğœ‰=0.(3.2) Substituting (2.4) into (3.2) to balance 𝑈3 with 𝑈𝜉𝜉, we find 𝑀=𝑁=1. Thus, the solution admits in the form𝑈(𝜉)=ğ‘Žâˆ’1𝑌−1(𝜉)+ğ‘Ž0+ğ‘Ž1𝑌(𝜉),(3.3) where ğ‘Žâˆ’1,ğ‘Ž0,ğ‘Ž1 are constants to be determined later.

Notice that𝑑(𝑠𝑛𝜉)𝑑𝜉=ğ‘ ğ‘›î…žğœ‰=𝑐𝑛𝜉𝑑𝑛𝜉,𝑑(𝑐𝑛𝜉)𝑑𝑑𝜉=𝑐𝑛′𝜉=−𝑠𝑛𝜉𝑑𝑛𝜉,(𝑑𝑛𝜉)𝑑𝜉=ğ‘‘ğ‘›î…žğœ‰=−𝑚𝑠𝑛𝜉𝑐𝑛𝜉,𝑐𝑛2𝜉=1−𝑠𝑛2𝜉,𝑑𝑛2𝜉=1−𝑚𝑠𝑛2𝜉.(3.4)

3.1. The Case of 𝑌=𝑌(𝜉)=𝑠𝑛𝜉=𝑠𝑛(𝜉,𝑚)

Substituting 𝑌=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 ğ‘Žâˆ’1,ğ‘Ž0,ğ‘Ž1, and 𝜆 of the following form:2𝜆2𝑟𝜇2î€¸ğ‘Žâˆ’ğ‘ğœŽâˆ’1+ğ‘Žğ‘Ž3−1=0,3ğ‘Žğ‘Ž2−1ğ‘Ž0=0,âˆ’ğ‘Žâˆ’1(1+𝑚)𝑟𝜆2𝜇2âˆ’î€·âˆ’ğœŽ+𝑏(1+𝑚)𝜆2î€¸ğœŽâˆ’2ğ‘Žğ‘Ž20+3ğ‘Žğ‘Ž2−1ğ‘Ž1=0,âˆ’ğœŽğ‘Ž0+ğ‘Žğ‘Ž30+6ğ‘Žğ‘Žâˆ’1ğ‘Ž0ğ‘Ž1=0,(−1−𝑚)𝑟𝜆2𝜇2ğ‘Ž1+−1+𝑏(1+𝑚)𝜆2î€»ğœŽğ‘Ž1+3ğ‘Žğ‘Ž20ğ‘Ž1+3ğ‘Žğ‘Žâˆ’1ğ‘Ž21=0.(3.5) Solving the system of the algebraic equations with the aid of Mathematica we can distinguish two cases, namely the following.

Case 1. ğ‘Ž1=0,ğ‘Ž0=0,ğ‘Žâˆ’1=√2ğœŽâˆšâˆšğ‘Ž(1+𝑚),𝜆=ğœŽî”î€·(1+𝑚)ğ‘ğœŽâˆ’ğ‘Ÿğœ‡2.(3.6)

Case 2. ğ‘Ž1=âˆšğœŽ(1+𝑚)âˆšğ‘Ž,ğ‘Žâˆ’1=âˆšğœŽâˆšğ‘Ž(1+𝑚),ğ‘Ž0√=0,𝜆=ğœŽî”î€·2(1+𝑚)ğ‘ğœŽâˆ’ğ‘Ÿğœ‡2.(3.7)

Substituting (3.6), (3.7) into (3.3), respectively, yield, the following solutions of ZK-MEW equation:√𝑢(𝑥,𝑦,𝑡)=2ğœŽâˆšğ‘Ž(1+𝑚)𝑠𝑛−1âŽ¡âŽ¢âŽ¢âŽ¢âŽ£âˆšğœŽî”î€·(1+𝑚)ğ‘ğœŽâˆ’ğ‘Ÿğœ‡2⎤⎥⎥⎥⎦,√(𝑥+ğœ‡ğ‘¦âˆ’ğœŽğ‘¡),𝑚(3.8)𝑢(𝑥,𝑦,𝑡)=ğœŽ(1+𝑚)âˆšğ‘ŽâŽ¡âŽ¢âŽ¢âŽ¢âŽ£âˆšğ‘ ğ‘›ğœŽî”2(1+𝑚)ğ‘ğœŽâˆ’ğ‘Ÿğœ‡2(⎤⎥⎥⎥⎦+√𝑥+ğœ‡ğ‘¦âˆ’ğœŽğ‘¡),ğ‘šğœŽâˆšğ‘Ž(1+𝑚)𝑠𝑛−1âŽ¡âŽ¢âŽ¢âŽ¢âŽ£âˆšğœŽî”î€·2(1+𝑚)ğ‘ğœŽâˆ’ğ‘Ÿğœ‡2⎤⎥⎥⎥⎦.(𝑥+ğœ‡ğ‘¦âˆ’ğœŽğ‘¡),𝑚(3.9) 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,√𝑢(𝑥,𝑦,𝑡)=ğœŽâˆšğ‘ŽâŽ¡âŽ¢âŽ¢âŽ¢âŽ£âˆšcothğœŽî”2î€·ğ‘ğœŽâˆ’ğ‘Ÿğœ‡2⎤⎥⎥⎥⎦,(𝑥+ğœ‡ğ‘¦âˆ’ğœŽğ‘¡)(3.10a)√𝑢(𝑥,𝑦,𝑡)=2ğœŽâˆšğ‘Žsin−1îƒ¬âˆšğœŽâˆšğ‘ğœŽâˆ’ğ‘Ÿğœ‡2(𝑥+ğœ‡ğ‘¦âˆ’ğœŽğ‘¡),(3.10b)√𝑢(𝑥,𝑦,𝑡)=2ğœŽâˆšğ‘ŽâŽ¡âŽ¢âŽ¢âŽ¢âŽ£âˆštanhğœŽ2î”î€·ğ‘ğœŽâˆ’ğ‘Ÿğœ‡2⎤⎥⎥⎥⎦+√(𝑥+ğœ‡ğ‘¦âˆ’ğœŽğ‘¡)ğœŽâˆšâŽ¡âŽ¢âŽ¢âŽ¢âŽ£âˆš2ğ‘ŽcothğœŽ2î”î€·ğ‘ğœŽâˆ’ğ‘Ÿğœ‡2(⎤⎥⎥⎥⎦,𝑥+ğœ‡ğ‘¦âˆ’ğœŽğ‘¡)(3.11a)√𝑢(𝑥,𝑦,𝑡)=ğœŽâˆšğ‘ŽâŽ¡âŽ¢âŽ¢âŽ¢âŽ£âˆšsinğœŽî”2î€·ğ‘ğœŽâˆ’ğ‘Ÿğœ‡2⎤⎥⎥⎥⎦+√(𝑥+ğœ‡ğ‘¦âˆ’ğœŽğ‘¡)ğœŽâˆšğ‘Žsin−1âŽ¡âŽ¢âŽ¢âŽ¢âŽ£âˆšğœŽî”2î€·ğ‘ğœŽâˆ’ğ‘Ÿğœ‡2(⎤⎥⎥⎥⎦.𝑥+ğœ‡ğ‘¦âˆ’ğœŽğ‘¡)(3.11b)

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.

3.2. The Case of 𝑌=𝑌(𝜉)=𝑑𝑛𝜉=𝑑𝑛(𝜉,𝑚)

The analysis proceeds of this case is as for Section 3.1. 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 ğ‘Žâˆ’1,ğ‘Ž0,ğ‘Ž1 and 𝜆 of the following form:2(−1+𝑚)𝜆2𝑟𝜇2î€¸ğ‘Žâˆ’ğ‘ğœŽâˆ’1+ğ‘Žğ‘Ž3−1=0,3ğ‘Žğ‘Ž2−1ğ‘Ž0=0,âˆ’ğ‘Žâˆ’1(−2+𝑚)𝑟𝜆2𝜇2+ğœŽ+2𝑏𝜆2ğœŽâˆ’ğ‘ğ‘šğœ†2ğœŽâˆ’3ğ‘Žğ‘Ž20+3ğ‘Žğ‘Ž2−1ğ‘Ž1=0,âˆ’ğœŽğ‘Ž0+ğ‘Žğ‘Ž30+6ğ‘Žğ‘Žâˆ’1ğ‘Ž0ğ‘Ž1=0,(2−𝑚)𝑟𝜆2𝜇2ğ‘Ž1+−1+𝑏(−2+𝑚)𝜆2î€»ğœŽğ‘Ž1+3ğ‘Žğ‘Ž20ğ‘Ž1+3ğ‘Žğ‘Žâˆ’1ğ‘Ž21ğ‘Ž=0,12𝜆2−𝑟𝜇2+ğ‘ğœŽ+ğ‘Žğ‘Ž21=0.(3.12) Solving the system of the algebraic equations with the aid of Mathematica we can distinguish three cases, namely, The following.

Case 1. ğ‘Žâˆ’1=0,ğ‘Ž0=0,ğ‘Ž1=√2ğœŽâˆšâˆšğ‘Ž(2−𝑚),𝜆=ğœŽî”î€·(2−𝑚)𝑟𝜇2î€¸âˆ’ğ‘ğœŽ.(3.13)

Case 2. ğ‘Žâˆ’1=⎷2(1−𝑚)ğœŽğ‘Žî‚€âˆš2−61−𝑚−𝑚,ğ‘Ž0=0,ğ‘Ž1=⎷2ğœŽğ‘Žî‚€âˆš2−6,⎷1−𝑚−𝑚𝜆=ğœŽî‚€âˆš2−61âˆ’ğ‘šâˆ’ğ‘šğ‘ğœŽâˆ’ğ‘Ÿğœ‡2.(3.14)

Case 3. ğ‘Žâˆ’1=2(1−𝑚)ğœŽğ‘Ž(2−𝑚),ğ‘Ž0=0,ğ‘Ž1√=0,𝜆=ğœŽî”î€·(2−𝑚)𝑟𝜇2î€¸âˆ’ğ‘ğœŽ.(3.15)

Substituting (3.13), (3.14), and (3.15) into (3.3), respectively, yields the following solutions of ZK-MEW equation:√𝑢(𝑥,𝑦,𝑡)=2ğœŽâˆšâŽ¡âŽ¢âŽ¢âŽ¢âŽ£âˆšğ‘Ž(2−𝑚)ğ‘‘ğ‘›ğœŽî”î€·(2−𝑚)𝑟𝜇2î€¸âŽ¤âŽ¥âŽ¥âŽ¥âŽ¦âˆ’ğ‘ğœŽ(𝑥+ğœ‡ğ‘¦âˆ’ğœŽğ‘¡),𝑚,(3.16)î„¶î„µî„µâŽ·ğ‘¢(𝑥,𝑦,𝑡)=2(1−𝑚)ğœŽğ‘Žî‚€âˆš2−61−𝑚−𝑚𝑑𝑛−1âŽ¡âŽ¢âŽ¢âŽ£î„¶î„µî„µâŽ·ğœŽî‚€âˆš2−61âˆ’ğ‘šâˆ’ğ‘šğ‘ğœŽâˆ’ğ‘Ÿğœ‡2⎤⎥⎥⎦+⎷(𝑥+ğœ‡ğ‘¦âˆ’ğœŽğ‘¡),𝑚2ğœŽğ‘Žî‚€âˆš2−6⎡⎢⎢⎣⎷1âˆ’ğ‘šâˆ’ğ‘šğ‘‘ğ‘›ğœŽî‚€âˆš2−61âˆ’ğ‘šâˆ’ğ‘šğ‘ğœŽâˆ’ğ‘Ÿğœ‡2(⎤⎥⎥⎦,𝑥+ğœ‡ğ‘¦âˆ’ğœŽğ‘¡),𝑚(3.17)îƒŽğ‘¢(𝑥,𝑦,𝑡)=2(1−𝑚)ğœŽğ‘Ž(2−𝑚)𝑑𝑛−1âŽ¡âŽ¢âŽ¢âŽ¢âŽ£âˆšğœŽî”î€·(2−𝑚)𝑟𝜇2î€¸âŽ¤âŽ¥âŽ¥âŽ¥âŽ¦âˆ’ğ‘ğœŽ(𝑥+ğœ‡ğ‘¦âˆ’ğœŽğ‘¡),𝑚.(3.18) 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,√𝑢(𝑥,𝑦,𝑡)=2ğœŽâˆšğ‘ŽâŽ¡âŽ¢âŽ¢âŽ¢âŽ£âˆšsechğœŽî”Â±î€·ğ‘Ÿğœ‡2î€¸âŽ¤âŽ¥âŽ¥âŽ¥âŽ¦âˆ’ğ‘ğœŽ(𝑥+ğœ‡ğ‘¦âˆ’ğœŽğ‘¡).(3.19) 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.

3.3. The Case of 𝑌=𝑌(𝜉)=𝑐𝑛𝜉=𝑐𝑛(𝜉,𝑚)

The analysis proceeds of this case is as for Sections 3.1 and 3.2. Substituting 𝑌=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 ğ‘Žâˆ’1,ğ‘Ž0,ğ‘Ž1, and 𝜆 of the following form:−2(−1+𝑚)𝜆2𝑟𝜇2î€¸ğ‘Žâˆ’ğ‘ğœŽâˆ’1+ğ‘Žğ‘Ž3−1=0,3ğ‘Žğ‘Ž2−1ğ‘Ž0ğ‘Ž=0,−1(−1+2𝑚)𝑟𝜆2𝜇2âˆ’ğœŽ+ğ‘ğœŽ(1−2𝑚)𝜆2+3ğ‘Žğ‘Ž20+3ğ‘Žğ‘Ž2−1ğ‘Ž1=0,âˆ’ğœŽğ‘Ž0+ğ‘Žğ‘Ž30+6ğ‘Žğ‘Žâˆ’1ğ‘Ž0ğ‘Ž1=0,(−1+2𝑚)𝑟𝜆2𝜇2ğ‘Ž1âˆ’ğœŽğ‘Ž1+ğ‘ğœŽ(1−2𝑚)𝜆2ğ‘Ž1+3ğ‘Žğ‘Ž20ğ‘Ž1+3ğ‘Žğ‘Ž21ğ‘Žâˆ’1=0,3ğ‘Žğ‘Ž21ğ‘Ž0ğ‘Ž=0,12𝑚𝜆2−𝑟𝜇2+ğ‘ğœŽ+ğ‘Žğ‘Ž21=0.(3.20) Solving the system of the algebraic equations (3.20) with the aid of Mathematica, we can distinguish three cases, namely, the following.

Case 1. ğ‘Ž0=0,ğ‘Ž1=2ğ‘šğœŽğ‘Ž(−1+2𝑚),ğ‘Žâˆ’1√=0,𝜆=ğœŽî”î€·(−1+2𝑚)𝑟𝜇2î€¸âˆ’ğ‘ğœŽ.(3.21)

Case 2. ğ‘Ž0=0,ğ‘Žâˆ’1=2(1−𝑚)ğœŽâˆ’ğ‘Ž(−1+2𝑚),ğ‘Ž1√=0,𝜆=ğœŽî”î€·(−1+2𝑚)𝑟𝜇2î€¸âˆ’ğ‘ğœŽ.(3.22)

Case 3. ğ‘Ž0=î‚™ğœŽğ‘Ž,ğ‘Ž1=−2ğ‘šğœŽğ‘Ž(1−2𝑚),ğ‘Žâˆ’1=0,𝜆=2ğœŽî€·(1−2𝑚)𝑟𝜇2î€¸âˆ’ğ‘ğœŽ.(3.23)

Substituting (3.21), (3.22), and (3.23) into (3.3), respectively, yield the following solutions of ZK-MEW equation:𝑢(𝑥,𝑦,𝑡)=2ğ‘šğœŽâŽ¡âŽ¢âŽ¢âŽ¢âŽ£âˆšğ‘Ž(−1+2𝑚)ğ‘ğ‘›ğœŽî”î€·(−1+2𝑚)𝑟𝜇2î€¸âŽ¤âŽ¥âŽ¥âŽ¥âŽ¦âˆ’ğ‘ğœŽ(𝑥+ğœ‡ğ‘¦âˆ’ğœŽğ‘¡),𝑚,(3.24)îƒŽğ‘¢(𝑥,𝑦,𝑡)=2(1−𝑚)ğœŽâˆ’ğ‘Ž(−1+2𝑚)𝑐𝑛−1âŽ¡âŽ¢âŽ¢âŽ¢âŽ£âˆšğœŽî”î€·(−1+2𝑚)𝑟𝜇2î€¸âŽ¤âŽ¥âŽ¥âŽ¥âŽ¦âˆ’ğ‘ğœŽ(𝑥+ğœ‡ğ‘¦âˆ’ğœŽğ‘¡),𝑚,(3.25)𝑢(𝑥,𝑦,𝑡)=ğœŽğ‘Žî‚™âˆ’2ğ‘šğœŽîƒ¬îƒŽğ‘Ž(1−2𝑚)𝑐𝑛2ğœŽî€·(1−2𝑚)𝑟𝜇2î€¸îƒ­âˆ’ğ‘ğœŽ(𝑥+ğœ‡ğ‘¦âˆ’ğœŽğ‘¡),𝑚.(3.26) 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)𝑢(𝑥,𝑦,𝑡)=2ğœŽğ‘Žîƒ¬âˆšsechğœŽâˆšğ‘Ÿğœ‡2îƒ­âˆ’ğ‘ğœŽ(𝑥+ğœ‡ğ‘¦âˆ’ğœŽğ‘¡),(3.27)𝑢(𝑥,𝑦,𝑡)=2ğœŽğ‘Žcos−1îƒ¬âˆšğœŽâˆšâˆ’ğ‘Ÿğœ‡2+ğ‘ğœŽ(𝑥+ğœ‡ğ‘¦âˆ’ğœŽğ‘¡).(3.28)

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.

Acknowledgments

The author would like to thank the anonymous referees for helpful suggestions that served to improve the paper. This paper is supported by Jiaying University (2011KJZ01).

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Copyright © 2011 Weimin Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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