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

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 𝑢_𝑡𝑢+𝑎𝑢𝑢_𝑥+𝑢_𝑥𝑥𝑥_𝑥+𝑢_𝑦𝑦=0,(1.1)_𝑡+𝑎𝑢𝑢_𝑥+∇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 𝑢_𝑡+𝑢_𝑥+12𝑢²_𝑥−𝑢_𝑥𝑥𝑡=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 by𝑢_𝑡+3𝑢²𝑢_𝑥−𝛽𝑢_𝑥𝑥𝑡=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 𝑢_𝑡+𝑎𝑢³_𝑥+𝑏𝑢_𝑥𝑡+𝑟𝑢_𝑦𝑦_𝑥=0,(1.5) 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 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,𝑢_𝑥,𝑢_𝑡,𝑢_𝑥𝑥,𝑢_𝑥𝑡,𝑢_𝑡𝑡,…=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,𝑈_𝜉,𝑈_𝜉𝜉,𝑈_𝜉𝜉𝜉,…=0.(2.3) By the extended Jacobi elliptic function expansion method, introduce the following ansatz 𝑈(𝜉)=^𝑁_𝑗=_−𝑀𝑎_𝑗𝑌^𝑗(𝜉),(2.4) where 𝑀,𝑁,𝑎_𝑗(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 𝑎_𝑗(𝑗=−𝑀,…,𝑁). Solving the above-mentioned equations with the Mathematica Software, then 𝑎_𝑗(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−𝜎𝑈_𝜉+𝑎𝑈³_𝜉+𝑟𝜆²𝜇²−𝑏𝜆²𝜎𝑈_𝜉𝜉𝜉=0.(3.1) Integrating (3.1) and setting the constant of integration to zero, we obtain−𝜎𝑈+𝑎𝑈³+𝑟𝜆²𝜇²−𝑏𝜆²𝜎𝑈_𝜉𝜉=0.(3.2) Substituting (2.4) into (3.2) to balance 𝑈³ with 𝑈_𝜉𝜉, we find 𝑀=𝑁=1. Thus, the solution admits in the form𝑈(𝜉)=𝑎₋₁𝑌⁻¹(𝜉)+𝑎₀+𝑎₁𝑌(𝜉),(3.3) where 𝑎₋₁,𝑎₀,𝑎₁ are constants to be determined later.

Notice that𝑑(𝑠𝑛𝜉)𝑑𝜉=𝑠𝑛^𝜉=𝑐𝑛𝜉𝑑𝑛𝜉,𝑑(𝑐𝑛𝜉)𝑑(𝑑𝑛𝜉)𝑑𝜉=𝑐𝑛′𝜉=−𝑠𝑛𝜉𝑑𝑛𝜉,𝑑𝜉=𝑑𝑛^𝑐𝑛₂𝜉=−𝑚𝑠𝑛𝜉𝑐𝑛𝜉,𝜉=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 𝑎₋₁,𝑎₀,𝑎₁, and 𝜆 of the following form:2𝜆²𝑟𝜇²−3𝑎𝑏𝑎𝜎₂₋₁𝑎₋₁+𝑎𝑎³⁻¹=0,−𝑎₋₁𝑎₀=0,(1+𝑚)𝑟𝜆²𝜇²−−𝜎𝑎−₀𝜎+𝑏(1+𝑚)𝜆²𝜎−2𝑎𝑎²⁰+3𝑎𝑎²⁻¹𝑎₁=0,(−1−𝑚)𝑟𝜆₂+𝑎𝑎³⁰+6𝑎𝑎₋₁𝑎₀𝑎₁=0,𝜇²𝑎₁+−1+𝑏(1+𝑚)𝜆²𝜎𝑎₁+3𝑎𝑎²⁰𝑎₁+3𝑎𝑎₋₁𝑎²¹=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. 𝑎₁=0,𝑎₀=0,𝑎₋₁=√2𝜎√𝑎(1+𝑚),𝜆=√𝜎(1+𝑚)𝑏𝜎−𝑟𝜇².(3.6)

Case 2. 𝑎₁=√𝜎(1+𝑚)√𝑎,𝑎⁻¹=√𝜎√𝑎(1+𝑚),𝑎⁰=0,𝜆=√𝜎2(1+𝑚)𝑏𝜎−𝑟𝜇².(3.7)

Substituting (3.6), (3.7) into (3.3), respectively, yield, the following solutions of ZK-MEW equation:𝑢(𝑥,𝑦,𝑡)=√2𝜎√𝑎(1+𝑚)𝑠𝑛⁻¹⎡⎢⎢⎢⎣√𝜎𝑢(𝑥,𝑦,𝑡)=√(1+𝑚)𝑏𝜎−𝑟𝜇²(𝑥+𝜇𝑦−𝜎𝑡),𝑚⎤⎥⎥⎥⎦,(3.8)𝜎(1+𝑚)√𝑎𝑠𝑛⎡⎢⎢⎢⎣√𝜎+√2(1+𝑚)𝑏𝜎−𝑟𝜇²(𝑥+𝜇𝑦−𝜎𝑡),𝑚⎤⎥⎥⎥⎦𝜎√𝑎(1+𝑚)𝑠𝑛⁻¹⎡⎢⎢⎢⎣√𝜎2(1+𝑚)𝑏𝜎−𝑟𝜇²(𝑥+𝜇𝑦−𝜎𝑡),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)=√𝜎√𝑎coth⎡⎢⎢⎢⎣√𝜎2𝑏𝜎−𝑟𝜇²(𝑥+𝜇𝑦−𝜎𝑡)⎤⎥⎥⎥⎦,(3.10a)𝑢(𝑥,𝑦,𝑡)=√2𝜎√𝑎sin⁻¹√𝜎√𝑏𝜎−𝑟𝜇²(𝑥+𝜇𝑦−𝜎𝑡),(3.10b)𝑢(𝑥,𝑦,𝑡)=√2𝜎√𝑎tanh⎡⎢⎢⎢⎣√𝜎+√2𝑏𝜎−𝑟𝜇²(𝑥+𝜇𝑦−𝜎𝑡)⎤⎥⎥⎥⎦𝜎√2𝑎coth⎡⎢⎢⎢⎣√𝜎2𝑏𝜎−𝑟𝜇²(𝑥+𝜇𝑦−𝜎𝑡)⎤⎥⎥⎥⎦,(3.11a)𝑢(𝑥,𝑦,𝑡)=√𝜎√𝑎sin⎡⎢⎢⎢⎣√𝜎+√2𝑏𝜎−𝑟𝜇²(𝑥+𝜇𝑦−𝜎𝑡)⎤⎥⎥⎥⎦𝜎√𝑎sin⁻¹⎡⎢⎢⎢⎣√𝜎2𝑏𝜎−𝑟𝜇²(𝑥+𝜇𝑦−𝜎𝑡)⎤⎥⎥⎥⎦.(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.

(a)

(b)

Figure 1 

(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)

Figure 2 

(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).

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 𝑎₋₁,𝑎₀,𝑎₁ and 𝜆 of the following form:2(−1+𝑚)𝜆²3𝑟𝑎𝜇𝑎²₂₋₁−𝑏𝜎𝑎₋₁+𝑎𝑎³⁻¹=0,−𝑎₋₁𝑎₀=0,(−2+𝑚)𝑟𝜆²𝜇−²𝜎+𝑎₀𝜎+2𝑏𝜆²𝜎−𝑏𝑚𝜆²𝜎−3𝑎𝑎²⁰+3𝑎𝑎²⁻¹𝑎₁=0,(2−𝑚)𝑟𝜆₂+𝑎𝑎³⁰+6𝑎𝑎₋₁𝑎₀𝑎₁=0,𝜇²𝑎₁+𝑎−1+𝑏(−2+𝑚)𝜆²𝜎𝑎₁+3𝑎𝑎²⁰𝑎₁+3𝑎𝑎₋₁𝑎²¹=0,₁2𝜆²−𝑟𝜇²+𝑏𝜎+𝑎𝑎²¹=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. 𝑎₋₁=0,𝑎₀=0,𝑎₁=√2𝜎√𝑎(2−𝑚),𝜆=√𝜎(2−𝑚)𝑟𝜇²−𝑏𝜎.(3.13)

Case 2. 𝑎₋₁=⎷2(1−𝑚)𝜎𝑎2−6√1−𝑚−𝑚,𝑎⁰=0,𝑎₁=⎷2𝜎𝜆=⎷𝑎2−6√1−𝑚−𝑚,𝜎2−6√1−𝑚−𝑚𝑏𝜎−𝑟𝜇².(3.14)

Case 3. 𝑎₋₁=2(1−𝑚)𝜎𝑎(2−𝑚),𝑎⁰=0,𝑎₁=0,𝜆=√𝜎(2−𝑚)𝑟𝜇²−𝑏𝜎.(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)=√2𝜎√𝑎(2−𝑚)𝑑𝑛⎡⎢⎢⎢⎣√𝜎(2−𝑚)𝑟𝜇²−𝑏𝜎(𝑥+𝜇𝑦−𝜎𝑡),𝑚⎤⎥⎥⎥⎦,(3.16)𝑢(𝑥,𝑦,𝑡)=⎷2(1−𝑚)𝜎𝑎2−6√1−𝑚−𝑚𝑑𝑛⁻¹⎡⎢⎢⎣⎷𝜎+⎷2−6√1−𝑚−𝑚𝑏𝜎−𝑟𝜇²(𝑥+𝜇𝑦−𝜎𝑡),𝑚⎤⎥⎥⎦2𝜎𝑎2−6√1−𝑚−𝑚𝑑𝑛⎡⎢⎢⎣⎷𝜎2−6√1−𝑚−𝑚𝑏𝜎−𝑟𝜇²(𝑥+𝜇𝑦−𝜎𝑡),𝑚⎤⎥⎥⎦,(3.17)𝑢(𝑥,𝑦,𝑡)=2(1−𝑚)𝜎𝑎(2−𝑚)𝑑𝑛⁻¹⎡⎢⎢⎢⎣√𝜎(2−𝑚)𝑟𝜇²−𝑏𝜎(𝑥+𝜇𝑦−𝜎𝑡),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)=√2𝜎√𝑎sech⎡⎢⎢⎢⎣√𝜎±𝑟𝜇²−𝑏𝜎(𝑥+𝜇𝑦−𝜎𝑡)⎤⎥⎥⎥⎦.(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.

(a)

(b)

Figure 3 

(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)

Figure 4 

(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).

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 𝑎₋₁,𝑎₀,𝑎₁, and 𝜆 of the following form:−2(−1+𝑚)𝜆²3𝑎𝑟𝑎𝜇²₂₋₁−𝑏𝜎𝑎₋₁+𝑎𝑎³⁻¹=0,𝑎𝑎₀=0,₋₁(−1+2𝑚)𝑟𝜆²−𝜇²𝜎𝑎−₀𝜎+𝑏𝜎(1−2𝑚)𝜆²+3𝑎𝑎²⁰+3𝑎𝑎²⁻¹𝑎₁=0,(−1+2𝑚)𝑟𝜆₂+𝑎𝑎³⁰+6𝑎𝑎₋₁𝑎₀𝑎₁=0,𝜇²𝑎₁−𝜎𝑎₁+𝑏3𝜎𝑎(𝑎1₂₁−2𝑚)𝜆²𝑎₁+3𝑎𝑎²⁰𝑎₁+3𝑎𝑎²¹𝑎₋₁=0,𝑎𝑎₀=0,₁2𝑚𝜆²−𝑟𝜇²+𝑏𝜎+𝑎𝑎²¹=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. 𝑎₀=0,𝑎₁=2𝑚𝜎𝑎(−1+2𝑚),𝑎⁻¹=0,𝜆=√𝜎(−1+2𝑚)𝑟𝜇²−𝑏𝜎.(3.21)

Case 2. 𝑎₀=0,𝑎₋₁=2(1−𝑚)𝜎−𝑎(−1+2𝑚),𝑎¹=0,𝜆=√𝜎(−1+2𝑚)𝑟𝜇²−𝑏𝜎.(3.22)

Case 3. 𝑎₀=𝜎𝑎,𝑎¹=−2𝑚𝜎𝑎(1−2𝑚),𝑎⁻¹=0,𝜆=2𝜎(1−2𝑚)𝑟𝜇²−𝑏𝜎.(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)=2𝑚𝜎𝑎(−1+2𝑚)𝑐𝑛⎡⎢⎢⎢⎣√𝜎(−1+2𝑚)𝑟𝜇²−𝑏𝜎(𝑥+𝜇𝑦−𝜎𝑡),𝑚⎤⎥⎥⎥⎦,(3.24)𝑢(𝑥,𝑦,𝑡)=2(1−𝑚)𝜎−𝑎(−1+2𝑚)𝑐𝑛⁻¹⎡⎢⎢⎢⎣√𝜎(−1+2𝑚)𝑟𝜇²−𝑏𝜎(𝑥+𝜇𝑦−𝜎𝑡),𝑚⎤⎥⎥⎥⎦,(3.25)𝑢(𝑥,𝑦,𝑡)=𝜎𝑎−2𝑚𝜎𝑎(1−2𝑚)𝑐𝑛2𝜎(1−2𝑚)𝑟𝜇²−𝑏𝜎(𝑥+𝜇𝑦−𝜎𝑡),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)=2𝜎𝑎sech√𝜎√𝑟𝜇²−𝑏𝜎(𝑥+𝜇𝑦−𝜎𝑡),(3.27)𝑢(𝑥,𝑦,𝑡)=2𝜎𝑎cos⁻¹√𝜎√−𝑟𝜇²+𝑏𝜎(𝑥+𝜇𝑦−𝜎𝑡).(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).