New Jacobi Elliptic Function Solutions for the Zakharov Equations

Abstract

A generalized (𝐺^′/𝐺)-expansion method is proposed to seek the exact solutions of nonlinear evolution equations. Being concise and straightforward, this method is applied to the Zakharov equations. As a result, some new Jacobi elliptic function solutions of the Zakharov equations are obtained. This method can also be applied to other nonlinear evolution equations in mathematical physics.

1. Introduction

In recent years, with the development of symbolic computation packages like Maple and Mathematica, searching for solutions of nonlinear differential equations directly has become more and more attractive [1–7]. This is because of the availability of computers symbolic system, which allows us to perform some complicated and tedious algebraic calculation and help us find new exact solutions of nonlinear differential equations.

In 2008, Wang et al. [8] introduced a new direct method called the (𝐺^′/𝐺)-expansion method to look for travelling wave solutions of nonlinear evolution equations (NLEEs). The method is based on the homogeneous balance principle and linear ordinary differential equation (LODE) theory. It is assumed that the traveling wave solutions can be expressed by a polynomial in (𝐺^′/𝐺), and that 𝐺=𝐺(𝜉) satisfies a second-order LODE 𝐺^′′+𝜆𝐺^′+𝜇𝐺=0. The degree of the polynomial can be determined by the homogeneous balance between the highest order derivative and nonlinear terms appearing in the given NPDEs. The coefficients of the polynomial can be obtained by solving a set of algebraic equations. Many literatures have shown that the (𝐺^′/𝐺)-expansion method is very effective, and many nonlinear equations have been successfully solved. Later, the further developed methods named the generalized (𝐺^′/𝐺)-expansion method [9], the modified (𝐺^′/𝐺)-expansion method [10], the extended (𝐺^′/𝐺)-expansion method [11], the improved (𝐺^′/𝐺)-expansion method [12], and the (𝐺^′/𝐺,1/𝐺)-expansion method [13] have been proposed.

As we know, when using the direct method, the choice of an appropriate auxiliary LODE is of great importance. In this paper, by introducing a new auxiliary LODE of different literature [8], we propose the generalized (𝐺^′/𝐺)-expansion method, which can be used to obtain travelling wave solutions of NLEEs.

In our contribution, we will seek exact solutions of the Zakharov equations [14]: 𝑛_𝑡𝑡−𝑐²_𝑠𝑛_𝑥𝑥=𝛽|𝐸|²_𝑥𝑥,𝑖𝐸_𝑡+𝛼𝐸_𝑥𝑥=𝛿𝑛𝐸, (1.1)(1.2) which are one of the classical models on governing the dynamics of nonlinear waves and describing the interactions between high- and low-frequency waves, where 𝑛 is the perturbed number density of the ion (in the low-frequency response), 𝐸 is the slow variation amplitude of the electric field intensity, 𝑐_𝑠 is the thermal transportation velocity of the electron ion, and 𝛼≠0, 𝛽≠0, 𝛿≠0 and 𝑐_𝑠 are constants.

Recently, many exact solutions of (1.1)-(1.2) have been successfully obtained by using the extended tanh-expansion method, the extended hyperbolic function method, the 𝐹-expansion method, the (𝐺^′/𝐺,1/𝐺)-expansion method [13–19], and so on.

In this paper, we construct the exact solutions to (1.1)-(1.2) by using the generalized (𝐺^′/𝐺)-expansion method. Furthermore, we show that the exact solutions are expressed by the Jacobi elliptic function.

2. The Generalized (𝐺^′/𝐺)-Expansion Method

Suppose that we have a nonlinear partial differential equation (PDE) for 𝑢(𝑥,𝑡) in the form 𝑁𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑡𝑡,𝑢_𝑥𝑡,𝑢_𝑥𝑥,…=0, (2.1) where 𝑁 is a polynomial in its arguments.

Step 1. By taking 𝑢(𝑥,𝑡)=𝑢(𝜉), 𝜉=𝑥−𝑐𝑡, we look for traveling wave solutions of (2.1) and transform it to the ordinary differential equation (ODE) 𝑁𝑢,−𝑐𝑢^′,𝑢^′,𝑐²𝑢^′′,−𝑐𝑢^′′,𝑢^′′,…=0. (2.2)

Step 2. Suppose the solution 𝑢 of (2.2) can be expressed as a finite series in the form 𝑢=𝑎₀+^𝑚_𝑖=1𝑎_𝑖𝑓^′𝑓^𝑖, (2.3) where 𝑎₀, 𝑎_𝑖(𝑖=1,2,…,𝑚) are constants to be determined later; 𝑓=𝑓(𝜉) is a solution of the auxiliary LODE 𝑓^′2=𝑃𝑓⁴+𝑄𝑓²+𝑅, (2.4) where 𝑃, 𝑄, and 𝑅 are constants.

Step 3. Determine the parameter 𝑚 by balancing the highest order nonlinear term and the highest order partial derivative of 𝑢 in (2.2).

Step 4. Substituting (2.3) and (2.4) into (2.2), setting all the coefficients of all terms with the same powers of (𝑓^′/𝑓)^𝑘(𝑘=1,2,…) to zero, we obtain a system of nonlinear algebraic equations (NAEs) with respect to the parameters 𝑐, 𝑎₀, 𝑎_𝑖(𝑖=1,2,…,𝑚). By solving the NAEs if available, we can determine those parameters explicitly.

Step 5. Assuming that the constants 𝑐, 𝑎₀, 𝑎_𝑖(𝑖=1,2,…,𝑚) can be obtained by solving the algebraic equations in Step 4, then substituting these constants and the known general solutions into (2.3), we can obtain the explicit solutions of (2.1) immediately.

3. Exact Solutions of the Zakharov Equations

In this section, we mainly apply the method proposed in Section 2 to seek the exact solutions of the Zakharov equations.

Since 𝐸(𝑥,𝑡) in (1.2) is a complex function and we are looking for the traveling wave solutions, thus we introduce a gauge transformation: 𝐸(𝑥,𝑡)=𝜑(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛=𝑛(𝜉),𝜉=𝑥−𝑐_𝑔𝑡+𝜉₁, (3.1) where 𝜑(𝑥,𝑡) is a real-valued function, 𝑐_𝑔, 𝑘, and 𝜔 are constants to be determined later, and 𝜉₀ and 𝜉₁ are constants. Substituting (3.1) into (1.1)-(1.2), we have 𝑐²_𝑔−𝑐²_𝑠𝑛^′′=𝛽𝜑^2′′,𝛼𝜑^′′+𝑖2𝛼𝑘−𝑐_𝑔𝜑^′+𝜔−𝛼𝑘²𝜑−𝛿𝑛𝜑=0. (3.2)(3.3)

Integrating (3.2) twice with respect to 𝜉, we have 𝛽𝑛=𝑐²_𝑔−𝑐²_𝑠𝜑²+𝐶₁𝜉+𝐶₂,𝑐²_𝑔−𝑐²_𝑠≠0, (3.4) where 𝐶₁ and 𝐶₂ are integration constants. Substituting (3.4) into (3.3), we have 𝛼𝜑^′′+𝑖2𝛼𝑘−𝑐_𝑔𝜑^′+𝜔−𝛼𝑘²𝛽𝜑−𝛿𝑐²_𝑔−𝑐²_𝑠𝜑²+𝐶₁𝜉+𝐶₂𝜑=0. (3.5)

In (3.5), we assume that 𝑐_𝑔=2𝛼𝑘,𝑘₁=𝛽𝑐²_𝑔−𝑐²_𝑠. (3.6)

Then (3.5) becomes the nonlinear ODE 𝛼𝜑^′′+𝜔−𝛼𝑘²𝜑−𝛿𝑘₁𝜑³+𝛿𝐶₁𝜉+𝐶₂𝜑=0. (3.7)

According to the homogeneous balance between 𝜑^′′ and 𝜑³ in (3.7), we obtain 𝑚=1. So we assume that 𝜑 can be expressed as 𝜑(𝜉)=𝑎₀+𝑎₁𝑓^′𝑓, (3.8) where 𝑓=𝑓(𝜉) satisfies (2.4). By using (2.4) and (3.8), it is easily derived that 𝜑^′′=2𝑎₁𝑓^′𝑓𝑓^′𝑓²−𝑄. (3.9)

Substituting (3.8) and (3.9) into (3.7), the left-hand side of (3.7) becomes a polynomial in (𝑓^′/𝑓) and 𝜉. Setting their coefficients to zero yields a system of algebraic equations in 𝑎₀, 𝑎₁, 𝑘, 𝑘₁, and 𝜔. Solving the overdetermined algebraic equations by Maple, we can obtain the following results: 𝑎₀=0,𝑎₁=2𝛼𝑘₁𝛿,𝜔=𝛼2𝑄+𝑘²+𝛿𝐶₂,𝐶₁=0. (3.10)

Substituting (3.10) into (3.8), we obtain 𝜑=2𝛼𝑘₁𝛿𝑓^′𝑓. (3.11)

Substituting (3.11) into (3.1) and (3.4), we have the following formal solution of (1.1)-(1.2): 𝐸=2𝛼𝑘₁𝛿𝑓^′𝑓𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛=2𝛼𝛿𝑓^′𝑓²+𝐶₂, (3.12) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(2𝑄+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

With the aid of the appendix [20] and from the formal solution (3.12), we get the following set of exact solutions of (1.1)-(1.2).

Case 1. Choosing 𝑃=𝑚², 𝑄=−(1+𝑚²), 𝑅=1, and 𝑓(𝜉)=sn(𝜉) and inserting them into (3.12), we obtain the Jacobi elliptic function solution of (1.1)-(1.2) 𝐸₁=2𝛼𝑘₁𝛿cs(𝜉)dn(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛₁=2𝛼𝛿cs²(𝜉)dn²(𝜉)+𝐶₂, (3.13) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(−2(1+𝑚²)+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 2. Choosing 𝑃=𝑚², 𝑄=−(1+𝑚²), 𝑅=1, and 𝑓(𝜉)=cd(𝜉) and inserting them into (3.12), we obtain the Jacobi elliptic function solution of (1.1)-(1.2) 𝐸₂=−2𝛼𝑘₁𝛿1−𝑚²sd(𝜉)nc(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛₂=2𝛼1−𝑚²²𝛿sd²(𝜉)nc²(𝜉)+𝐶₂, (3.14) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(−2(1+𝑚²)+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 3. Choosing 𝑃=−𝑚², 𝑄=2𝑚²−1, 𝑅=1−𝑚², and 𝑓(𝜉)=cn(𝜉), we obtain 𝐸₃=−2𝛼𝑘₁𝛿dc(𝜉)sn(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛₃=2𝛼𝛿dc²(𝜉)sn²(𝜉)+𝐶₂, (3.15) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(2(2𝑚²−1)+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 4. Choosing 𝑃=−1, 𝑄=2−𝑚², 𝑅=𝑚²−1, and 𝑓(𝜉)=dn(𝜉), we obtain 𝐸₄=−2𝛼𝑘₁𝛿𝑚²cd(𝜉)sn(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛₄=2𝛼𝑚⁴𝛿cd²(𝜉)sn²(𝜉)+𝐶₂, (3.16) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(2(2−𝑚²)+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 5. Choosing 𝑃=1, 𝑄=−(1+𝑚²), 𝑅=𝑚², and 𝑓(𝜉)=ns(𝜉), we obtain 𝐸₅=−2𝛼𝑘₁𝛿cs(𝜉)dn(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛₅=2𝛼𝛿cs²(𝜉)dn²(𝜉)+𝐶₂, (3.17) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(−2(1+𝑚²)+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 6. Choosing 𝑃=1, 𝑄=−(1+𝑚²), 𝑅=𝑚², and 𝑓(𝜉)=dc(𝜉), we obtain 𝐸₆=2𝛼𝑘₁𝛿1−𝑚²sc(𝜉)nd(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛₆=2𝛼1−𝑚²²𝛿sc²(𝜉)nd²(𝜉)+𝐶₂, (3.18) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(−2(1+𝑚²)+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 7. Choosing 𝑃=1−𝑚², 𝑄=2𝑚²−1, 𝑅=−𝑚², and 𝑓(𝜉)=nc(𝜉), we obtain 𝐸₇=2𝛼𝑘₁𝛿dc(𝜉)sn(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛₇=2𝛼𝛿dc²(𝜉)sn²(𝜉)+𝐶₂, (3.19) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(2(2𝑚²−1)+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 8. Choosing 𝑃=𝑚²−1, 𝑄=2−𝑚², 𝑅=−1, and 𝑓(𝜉)=nd(𝜉), we obtain 𝐸₈=2𝛼𝑘₁𝛿𝑚²cd(𝜉)sn(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛₈=2𝛼𝑚⁴𝛿cd²(𝜉)sn²(𝜉)+𝐶₂, (3.20) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(2(2−𝑚²)+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 9. Choosing 𝑃=1−𝑚², 𝑄=2−𝑚², 𝑅=1, and 𝑓(𝜉)=sc(𝜉), we obtain 𝐸₉=2𝛼𝑘₁𝛿dc(𝜉)ns(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛₉=2𝛼𝛿dc²(𝜉)ns²(𝜉)+𝐶₂, (3.21) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(2(2−𝑚²)+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 10. Choosing 𝑃=−𝑚²(1−𝑚²), 𝑄=2𝑚²−1, 𝑅=1, and 𝑓(𝜉)=sd(𝜉), we obtain 𝐸₁₀=2𝛼𝑘₁𝛿cd(𝜉)ns(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛₁₀=2𝛼𝛿cd²(𝜉)ns²(𝜉)+𝐶₂, (3.22) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(2(2𝑚²−1)+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 11. Choosing 𝑃=1, 𝑄=2−𝑚², 𝑅=1−𝑚², and 𝑓(𝜉)=cs(𝜉), we obtain 𝐸₁₁=2𝛼𝑘₁𝛿ds(𝜉)nc(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛₁₁=2𝛼𝛿ds²(𝜉)nc²(𝜉)+𝐶₂, (3.23) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(2(2−𝑚²)+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 12. Choosing 𝑃=1, 𝑄=2𝑚²−1, 𝑅=−𝑚²(1−𝑚²), and 𝑓(𝜉)=ds(𝜉), we obtain 𝐸₁₂=−2𝛼𝑘₁𝛿cs(𝜉)nd(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛₁₂=2𝛼𝛿cs²(𝜉)nd²(𝜉)+𝐶₂, (3.24) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(2(2𝑚²−1)+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 13. Choosing 𝑃=1/4, 𝑄=(1−2𝑚²)/2, 𝑅=1/4, and 𝑓(𝜉)=ns(𝜉)±cs(𝜉), we obtain 𝐸₁₃=∓2𝛼𝑘₁𝛿ds(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛₁₃=2𝛼𝛿ds²(𝜉)+𝐶₂, (3.25) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(1−2𝑚²+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 14. Choosing 𝑃=(1−𝑚²)/4, 𝑄=(1+𝑚²)/2, 𝑅=(1−𝑚²)/4, and 𝑓(𝜉)=nc(𝜉)±sc(𝜉), we obtain 𝐸₁₄=±2𝛼𝑘₁𝛿dc(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛₁₄=2𝛼𝛿dc²(𝜉)+𝐶₂, (3.26) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(1+𝑚²+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 15. Choosing 𝑃=1/4, 𝑄=(𝑚²−2)/2, 𝑅=𝑚²/4, and 𝑓(𝜉)=ns(𝜉)±ds(𝜉), we obtain 𝐸₁₅=∓2𝛼𝑘₁𝛿cs(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛₁₅=2𝛼𝛿cs²(𝜉)+𝐶₂, (3.27) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(𝑚²−2+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 16. Choosing 𝑃=𝑚²/4, 𝑄=(𝑚²−2)/2, 𝑅=𝑚²/4, and 𝑓(𝜉)=sn(𝜉)±𝑖cn(𝜉), we obtain 𝐸₁₆=2𝛼𝑘₁𝛿dn(𝜉)(cn(𝜉)∓𝑖sn(𝜉))𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]}(𝜉)(𝜉),𝑛sn±𝑖cn₁₆=2𝛼dn²(𝜉)(cn(𝜉)∓𝑖sn(𝜉))²𝛿(sn(𝜉)±𝑖cn(𝜉))²+𝐶₂, (3.28) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(𝑚²−2+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 17. Choosing 𝑃=𝑚²/4, 𝑄=(𝑚²−2)/2, 𝑅=𝑚²/4, and 𝑓(𝜉)=√𝑚²−1sd(𝜉)±cd(𝜉), we obtain 𝐸₁₇=2𝛼𝑘₁𝛿√𝑚²−1cn(𝜉)±𝑚²sn(𝜉)∓sn(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]}dn(𝜉)√𝑚²(𝜉)(𝜉,𝑛−1sn±cn)₁₇=2𝛼√𝑚²−1cn(𝜉)±𝑚²sn(𝜉)∓sn(𝜉)²𝛿dn²(𝜉)√𝑚²−1sn(𝜉)±cn(𝜉)²+𝐶₂, (3.29) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(𝑚²−2+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 18. Choosing 𝑃=1/4, 𝑄=(1−2𝑚²)/2, 𝑅=1/4, and 𝑓(𝜉)=𝑚cd(𝜉)±𝑖√1−𝑚²nd(𝜉), we obtain 𝐸₁₈=2𝛼𝑘₁𝛿𝑚sn(𝜉)±𝑚²∓1+𝑖𝑚√1−𝑚²cn(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]}dn(𝜉)𝑖√1−𝑚²(𝜉,𝑛±𝑚cn)₁₈=2𝛼𝑚²sn²(𝜉)±𝑚²∓1+𝑖𝑚√1−𝑚²cn(𝜉)²𝛿dn²(𝜉)𝑖√1−𝑚²±𝑚cn(𝜉)²+𝐶₂, (3.30) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(1−2𝑚²+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 19. Choosing 𝑃=1/4, 𝑄=(1−2𝑚²)/2, 𝑅=1/4, and 𝑓(𝜉)=𝑚sn(𝜉)±𝑖dn(𝜉), we obtain 𝐸₁₉=2𝛼𝑘₁𝛿𝑚cn(𝜉)(dn(𝜉)∓𝑖𝑚sn(𝜉))𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]}(𝜉)(𝜉),𝑛𝑚sn±𝑖dn₁₉=2𝛼𝑚²cn²(𝜉)(dn(𝜉)∓𝑖𝑚sn(𝜉))²𝛿(𝑚sn(𝜉)±𝑖dn(𝜉))²+𝐶₂, (3.31) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(1−2𝑚²+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 20. Choosing 𝑃=1/4, 𝑄=(1−2𝑚²)/2, 𝑅=1/4, and 𝑓(𝜉)=√𝑚²−1sc(𝜉)±𝑖dc(𝜉), we obtain 𝐸₂₀=2𝛼𝑘₁𝛿√𝑚²−1dn(𝜉)∓𝑖𝑚²sn(𝜉)±𝑖sn(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]}cn(𝜉)√𝑚²(𝜉)(𝜉,𝑛−1sn±𝑖dn)₂₀=2𝛼√𝑚²−1dn(𝜉)∓𝑖𝑚²sn(𝜉)±𝑖sn(𝜉)²𝛿cn²(𝜉)√𝑚²−1sn(𝜉)±𝑖dn(𝜉)²+𝐶₂, (3.32) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(1−2𝑚²+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 21. Choosing 𝑃=(𝑚²−1)/4, 𝑄=(𝑚²+1)/2, 𝑅=(𝑚²−1)/4, and 𝑓(𝜉)=𝑚sd(𝜉)±nd(𝜉), we obtain 𝐸₂₁=±2𝛼𝑘₁𝛿𝑚cd(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛₂₁=2𝛼𝑚²𝛿cd²(𝜉)+𝐶₂, (3.33) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(𝑚²+1+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 22. Choosing 𝑃=𝑚²/4, 𝑄=(𝑚²−2)/2, 𝑅=1/4, and 𝑓(𝜉)=sn(𝜉)/(1±dn(𝜉)), we obtain 𝐸₂₂=±2𝛼𝑘₁𝛿cs(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛₂₂=2𝛼𝛿cs²(𝜉)+𝐶₂, (3.34) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(𝑚²−2+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 23. Choosing 𝑃=−1/4, 𝑄=(𝑚²+1)/2, 𝑅=(1−𝑚²)²/4, and 𝑓(𝜉)=𝑚cn(𝜉)±dn(𝜉), we obtain 𝐸₂₃=∓2𝛼𝑘₁𝛿𝑚sn(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛₂₃=2𝛼𝑚²𝛿sn²(𝜉)+𝐶₂, (3.35) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(𝑚²+1+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 24. Choosing 𝑃=(1−𝑚²)²/4, 𝑄=(𝑚²+1)/2, 𝑅=1/4, and 𝑓(𝜉)=ds(𝜉)±cs(𝜉), we obtain 𝐸₂₄=∓2𝛼𝑘₁𝛿ns(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛₂₄=2𝛼𝛿ns²(𝜉)+𝐶₂, (3.36) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(𝑚²+1+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 25. Choosing 𝑃=1/4, 𝑄=(𝑚²−2)/2, 𝑅=𝑚⁴/4, and 𝑓(𝜉)=dc(𝜉)±√1−𝑚²nc(𝜉), we obtain 𝐸₂₅=2𝛼𝑘₁𝛿sn(𝜉)∓𝑚²±1+√1−𝑚²dn(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]}cn(𝜉)√1−𝑚²(𝜉,𝑛±dn)₂₅=2𝛼sn²(𝜉)∓𝑚²±1+√1−𝑚²dn(𝜉)²𝛿cn²(𝜉)√1−𝑚²±dn(𝜉)²+𝐶₂, (3.37) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(𝑚²−2+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 26. Choosing 𝑅=𝑚²𝑄²/(𝑚²+1)²𝑃, 𝑄<0, 𝑃>0, and 𝑓(𝜉)=−𝑚²𝑄/(𝑚²+1)𝑃sn((−𝑄/(𝑚²+1))𝜉), we obtain 𝐸₂₆=2𝛼𝑘₁𝛿−𝑄𝑚²+1cs−𝑄𝑚²𝜉+1dn−𝑄𝑚²𝜉𝑒+1^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛₂₆=−2𝛼𝑄𝛿𝑚²+1cs²−𝑄𝑚²𝜉+1dn²−𝑄𝑚²𝜉+1+𝐶₂, (3.38) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(2𝑄+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 27. Choosing 𝑅=(1−𝑚²)𝑄²/(𝑚²−2)²𝑃, 𝑄>0, 𝑃<0, and 𝑓(𝜉)=−𝑄/(2−𝑚²)𝑃dn((𝑄/(2−𝑚²))𝜉), we obtain 𝐸₂₇=−2𝛼𝑘₁𝛿𝑄2−𝑚²𝑚²cd𝑄2−𝑚²𝜉sn𝑄2−𝑚²𝜉𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛₂₇=2𝛼𝑄𝑚⁴𝛿2−𝑚²cd²𝑄2−𝑚²𝜉sn²𝑄2−𝑚²𝜉+𝐶₂, (3.39) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(2𝑄+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 28. Choosing 𝑅=𝑚²(𝑚²−1)𝑄²/(2𝑚²−1)²𝑃, 𝑄>0, 𝑃<0, and 𝑓(𝜉)=−(𝑚²𝑄/(2𝑚²−1)𝑃)cn((𝑄/(2𝑚²−1))𝜉), we obtain 𝐸₂₈=−2𝛼𝑘₁𝛿𝑄2𝑚²−1dc𝑄2𝑚²𝜉−1sn𝑄2𝑚²𝜉𝑒−1^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛₂₈=2𝛼𝑄𝛿2𝑚²−1dc²𝑄2𝑚²𝜉−1sn²𝑄2𝑚²𝜉−1+𝐶₂, (3.40) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(2𝑄+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 29. Choosing 𝑃=1, 𝑄=2−4𝑚², 𝑅=1, and 𝑓(𝜉)=sn(𝜉)dn(𝜉)/cn(𝜉), we obtain 𝐸₂₉=2𝛼𝑘₁𝛿1−2𝑚²sn²(𝜉)+𝑚²sn⁴(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]}(𝜉)(𝜉)(𝜉),𝑛cnsndn₂₉=2𝛼1−2𝑚²sn²(𝜉)+𝑚²sn⁴(𝜉)²²𝛿cn²(𝜉)sn²(𝜉)dn²(𝜉)+𝐶₂, (3.41) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(2(2−4𝑚²)+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 30. Choosing 𝑃=𝑚⁴, 𝑄=2𝑚²−4, 𝑅=1, and 𝑓(𝜉)=sn(𝜉)cn(𝜉)/dn(𝜉), we obtain 𝐸₃₀=2𝛼𝑘₁𝛿1−2sn²(𝜉)+𝑚²sn⁴(𝜉)𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]}(𝜉)(𝜉)(𝜉),𝑛dnsncn₃₀=2𝛼1−2sn²(𝜉)+𝑚²sn⁴(𝜉)²𝛿dn²(𝜉)sn²(𝜉)cn²(𝜉)+𝐶₂, (3.42) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(2(2𝑚²−4)+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 31. Choosing 𝑃=1, 𝑄=2𝑚²+2, 𝑅=1−2𝑚²+𝑚⁴, and 𝑓(𝜉)=cn(𝜉)dn(𝜉)/sn(𝜉), we obtain 𝐸₃₁=2𝛼𝑘₁𝛿𝑚²sn⁴(𝜉)−1𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]}(𝜉)(𝜉)(𝜉),𝑛sncndn₃₁=2𝛼𝑚²sn⁴(𝜉)−1²𝛿sn²(𝜉)cn²(𝜉)dn²(𝜉)+𝐶₂, (3.43) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(2(2𝑚²+2)+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 32. Choosing 𝑃=𝐴²(𝑚−1)²/4, 𝑄=(𝑚²+1)/2+3𝑚, 𝑅=(𝑚−1)²/4𝐴², and 𝑓(𝜉)=dn(𝜉)cn(𝜉)/𝐴(1+sn(𝜉))(1+𝑚sn(𝜉)), we obtain 𝐸₃₂=2𝛼𝑘₁𝛿𝑚²sn²(𝜉)+𝑚sn²(𝜉)−1−𝑚𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]}(𝜉)(𝜉),𝑛dncn₃₂=2𝛼𝑚²sn²(𝜉)+𝑚sn²(𝜉)−1−𝑚²𝛿dn²(𝜉)cn²(𝜉)+𝐶₂, (3.44) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(𝑚²+1+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 33. Choosing 𝑃=𝐴²(𝑚+1)²/4, 𝑄=(𝑚²+1)/2−3𝑚, 𝑅=(𝑚+1)²/4𝐴², and 𝑓(𝜉)=dn(𝜉)cn(𝜉)/𝐴(1+sn(𝜉))(1−𝑚sn(𝜉)), we obtain 𝐸₃₃=2𝛼𝑘₁𝛿𝑚²sn²(𝜉)−𝑚sn²(𝜉)+𝑚−1𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]}(𝜉)(𝜉),𝑛dncn₃₃=2𝛼𝑚²sn²(𝜉)−𝑚sn²(𝜉)+𝑚−1²𝛿dn²(𝜉)cn²(𝜉)+𝐶₂, (3.45) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(𝑚²+1+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 34. Choosing 𝑃=−4/𝑚, 𝑄=6𝑚−𝑚²−1, 𝑅=−2𝑚³+𝑚⁴+𝑚², and 𝑓(𝜉)=𝑚cn(𝜉)dn(𝜉)/(𝑚sn²(𝜉)+1), we obtain 𝐸₃₄=2𝛼𝑘₁𝛿sn(𝜉)𝑚sn²(𝜉)−1+2𝑚²sn²(𝜉)+𝑚³sn²(𝜉)−𝑚²−2𝑚𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]}𝑚sn²(𝜉)(𝜉)(𝜉),𝑛+1cndn₃₄=2𝛼sn²(𝜉)𝑚sn²(𝜉)−1+2𝑚²sn²(𝜉)+𝑚³sn²(𝜉)−𝑚²−2𝑚²𝛿𝑚sn²(𝜉)+1²cn²(𝜉)dn²(𝜉)+𝐶₂, (3.46) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(2(6𝑚−𝑚²−1)+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 35. Choosing 𝑃=4/𝑚, 𝑄=−6𝑚−𝑚²−1, 𝑅=2𝑚³+𝑚⁴+𝑚², and 𝑓(𝜉)=𝑚cn(𝜉)dn(𝜉)/(𝑚sn²(𝜉)−1), we obtain 𝐸₃₅=2𝛼𝑘₁𝛿sn(𝜉)𝑚sn²(𝜉)+1−2𝑚²sn²(𝜉)+𝑚³sn²(𝜉)+𝑚²−2𝑚𝑒^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]}𝑚sn²(𝜉)(𝜉)(𝜉),𝑛−1cndn₃₅=2𝛼sn²(𝜉)𝑚sn²(𝜉)+1−2𝑚²sn²(𝜉)+𝑚³sn²(𝜉)+𝑚²−2𝑚²𝛿𝑚sn²(𝜉)−1²cn²(𝜉)dn²(𝜉)+𝐶₂, (3.47) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(−2(6𝑚+𝑚²+1)+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 36. Choosing 𝑃=−(𝑚²+2𝑚+1)𝐵², 𝑄=2𝑚²+2, 𝑅=(2𝑚−𝑚²−1)/𝐵², and 𝑓(𝜉)=(𝑚sn²(𝜉)−1)/𝐵(𝑚sn²(𝜉)+1), we obtain 𝐸₃₆=42𝛼𝑘₁𝛿𝑚sn(𝜉)cn(𝜉)dn(𝜉)𝑚²sn⁴(𝜉)𝑒−1^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛₃₆=32𝛼𝑚²sn²(𝜉)cn²(𝜉)dn²(𝜉)𝛿𝑚²sn⁴(𝜉)−1²+𝐶₂, (3.48) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(2(2𝑚²+2)+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

Case 37. Choosing 𝑃=−(𝑚²−2𝑚+1)𝐵², 𝑄=2𝑚²+2, 𝑅=−(2𝑚+𝑚²+1)/𝐵², and 𝑓(𝜉)=(𝑚sn²(𝜉)+1)/𝐵(𝑚sn²(𝜉)−1), we obtain 𝐸₃₇=−42𝛼𝑘₁𝛿𝑚sn(𝜉)cn(𝜉)dn(𝜉)𝑚²sn⁴(𝜉)𝑒−1^{[𝑖(𝑘𝑥−𝜔𝑡+𝜉0)]},𝑛₃₇=32𝛼𝑚²sn²(𝜉)cn²(𝜉)dn²(𝜉)𝛿𝑚²sn⁴(𝜉)−1²+𝐶₂, (3.49) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁, 𝜔=𝛼(2(2𝑚²+2)+𝑘²)+𝛿𝐶₂, and 𝑘₁=𝛽/(4𝛼²𝑘²−𝑐²_𝑠).

4. Conclusions

In this paper, by using the generalized (𝐺^′/𝐺)-expansion method, we have successfully obtained some exact solutions of Jacobi elliptic function form of the Zakharov equations. When the modulus of the Jacobi elliptic function 𝑚→0 or 1, the corresponding solitary wave solutions and trigonometric function solutions are also obtained. This work shows that the generalized (𝐺^′/𝐺)-expansion method provides a very effective and powerful tool for solving nonlinear equations in mathematical physics.

Appendix

For more details see Table 1 and [20].

Table 1 

Relations between the coefficients (𝑃, 𝑄, and 𝑅) and corresponding 𝑓(𝜉) in 𝑓^′2=𝑃𝑓⁴+𝑄𝑓²+𝑅.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11161020), the Natural Science Foundation of Yunnan Province (2011FZ193), the Natural Science Foundation of Education Committee of Yunnan Province (2012Y452), and the Scientific Foundation of Kunming University (XJ11L021).