A New Extended Jacobi Elliptic Function Expansion Method and Its Application to the Generalized Shallow Water Wave Equation

Abstract

With the aid of symbolic computation, a new extended Jacobi elliptic function expansion method is presented by means of a new ansatz, in which periodic solutions of nonlinear evolution equations, which can be expressed as a finite Laurent series of some 12 Jacobi elliptic functions, are very effective to uniformly construct more new exact periodic solutions in terms of Jacobi elliptic function solutions of nonlinear partial differential equations. As an application of the method, we choose the generalized shallow water wave (GSWW) equation to illustrate the method. As a result, we can successfully obtain more new solutions. Of course, more shock wave solutions or solitary wave solutions can be gotten at their limit condition.

1. Introduction

In recent years, the nonlinear partial differential equations (NPDEs) are widely used to describe many important phenomena and dynamic processes in physics, mechanics, chemistry, biology, and so forth. With the development of soliton theory, There has been a great amount of activities aiming to find methods for exact solutions of nonlinear differential equations, such as Bäcklund transformation, Darboux transformation, Cole-Hopf transformation, similarity reduction method, variable separation approach, Exp-function method, homogeneous balance method, varied tanh methods, and varied Jacobi elliptic function methods [1–21].

Among those, the direct ansätz method [8–21] provides a straightforward and effective algorithm to obtain such particular solutions for a large number of nonlinear partial differential equations, in which the starting point is the ansätz that the solution sought is expressible as a finite series of special function, such as tanh function, sech function, tan function, sec function, sine-cosine function, Weierstrass elliptic function, theta function, and Jacobi elliptic function.

In this paper, a new Jacobi elliptic function expansion method is presented by means of a new general ansätz and is more powerful to uniformly construct more new exact doubly-periodic solutions in terms of Jacobi elliptic functions of nonlinear partial differential equations. The algorithm and its application are demonstrated later.

This paper is organized as follows. In Section 2, we summarize the extended Jacobi elliptic function expansion method. In Section 3, we apply the extended method to the generalized shallow water wave (GSWW) equation and bring out more new solutions. Conclusions will be presented in finally.

2. Summary of the Extended Jacobi Elliptic Function Expansion Method

In the following, we would like to outline the main steps of our extended method.

Step 1. For a given nonlinear partial differential equation with some physical fields 𝑢(𝑥,𝑦,𝑡) in three variables 𝑥, 𝑦, 𝑡, 𝐹𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑦,𝑢_𝑡𝑡,𝑢_𝑥𝑡,𝑢_𝑦𝑡,𝑢_𝑥𝑥,𝑢_𝑦𝑦,𝑢_𝑥𝑦,…=0, (2.1) by using the wave transformation 𝑢𝑥,𝑦,𝑡=𝑢(𝜉),𝜉=𝑘𝑥+𝑙𝑦−𝑐𝑡, (2.2) where 𝑘, 𝑙, and 𝑐 are constants to be determined later. Then the nonlinear partial differential equation (2.1) is reduced to a nonlinear ordinary differential equation (ODE): 𝐺𝑢,𝑢^′,𝑢^′′,…=0. (2.3)

Step 2. We introduce some new ansätz in terms of finite Jacobi elliptic function expansion in the following forms:(1)sn^′𝜉/sn𝜉 expansion:𝑢=𝑎₀+^𝑛_𝑖=1𝑎_𝑖sn^′(𝜉)sn(𝜉)^𝑖+𝑏_𝑖sn^′(𝜉)sn(𝜉)^−𝑖, (2.4a)(2)cn^′𝜉/cn𝜉 expansion: 𝑢=𝑎₀+^𝑛_𝑖=1𝑎_𝑖cn^′(𝜉)cn(𝜉)^𝑖+𝑏_𝑖cn^′(𝜉)cn(𝜉)^−𝑖, (2.4b)(3)dn^′𝜉/dn𝜉 expansion: 𝑢=𝑎₀+^𝑛_𝑖=1𝑎_𝑖dn^′(𝜉)dn(𝜉)^𝑖+𝑏_𝑖dn^′(𝜉)dn(𝜉)^−𝑖, (2.4c)where sn𝜉, cn𝜉, dn𝜉 are Jacobi elliptic sine function, the Jacobi elliptic cosine function, and the Jacobi elliptic function of the third kind, which are periodic and possess the following properties:

(1) properties of triangular functions cn²𝜉+sn²𝜉=dn²𝜉+𝑚²sn²𝜉=1, (2.5)

(2) derivatives of the Jacobi elliptic functions sn^′𝜉=cn𝜉dn𝜉,cn^′𝜉=−sn𝜉dn𝜉,dn^′𝜉=−𝑚²sn𝜉cn𝜉, (2.6) where 𝑚 is a modulus. The Jacobi-Glaisher functions for elliptic function can be found in [22, 23]. It is necessary to point out that above combinations only require solving the recurrent coefficient relation or derivative relation for the terms of polynomial for computation closed. Therefore, other Jacobi elliptic functions can be chosen to combine as new ansätz. For simplicity, we omit them here.

Step 3. In order to obtain the value of 𝑛 in (2.4a)–(2.4c), we define the degree of 𝑢(𝜉) as 𝐷[𝑢(𝜉)]=𝑛, which gives rise to the degree of other expressions as 𝐷𝑢^(𝛼)=𝑛+𝛼,𝐷𝑢^𝛽𝑢^(𝛼)𝑠=𝑛𝛽+(𝛼+𝑛)𝑠. (2.7) Therefore, we can get the value of 𝑛 in (2.4a)–(2.4c). If 𝑛 is a nonnegative integer, then we first make the transformation 𝑢=𝜔^𝑛.

Step 4. Respectively substitute three cases of (2.4a)–(2.4c) into (2.3) along with (2.5) and (2.6) and then respectively set all coefficients of sn^𝑖𝜉cn^𝑗𝜉dn^𝑘𝜉(𝑖=0,1,2,…;𝑗=0,1;𝑘=0,1) to be zero to get an overdetermined system of nonlinear algebraic equations with respect to 𝑎₀, 𝑎_𝑖, 𝑏_𝑖, 𝑘, 𝑙, 𝑐(𝑖=0,1,2,…).

Step 5. By use of the Maple software package “Charsets” by Dongming Wang, which is based on the Wu-elimination method [24], solving the overdetermined algebraic equations, we would end up with the explicit expressions for 𝑎₀, 𝑎_𝑖, 𝑏_𝑖, 𝑘, 𝑙, 𝑐(𝑖=0,1,2,…). In this way, we can get periodic solutions with Jacobi elliptic function.
Since lim_𝑚→1sn𝜉=tanh𝜉,lim_𝑚→1cn𝜉=sech𝜉,lim_𝑚→1dn𝜉=sech𝜉,lim_𝑚→0sn𝜉=sin𝜉,lim_𝑚→0cn𝜉=cos𝜉,lim_𝑚→0dn𝜉=1, (2.8)𝑢 degenerate respectively as in the following form:

(1) solitary wave solutions:𝑢(𝜉)=𝑎₀+^𝑛_𝑖=1𝑎_𝑖sech^𝑖𝜉csch^𝑖𝜉+𝑏_𝑖cosh^𝑖𝜉sinh^𝑖𝑢(𝜉)𝜉,=𝑎₀+^𝑛_𝑖=1𝑎_𝑖(−tanh𝜉)^𝑖+𝑏_𝑖(−coth𝜉)^𝑖, (2.9a)(2.9b)

(2) triangular function formal solutions:𝑢(𝜉)=𝑎₀+^𝑛_𝑖=1𝑎_𝑖cot^𝑖𝜉+𝑏_𝑖tan^𝑖𝑢(𝜉)𝜉,=𝑎₀+^𝑛_𝑖=1𝑎_𝑖(−tan𝜉)^𝑖+𝑏_𝑖(−cot𝜉)^𝑖. (2.10a)(2.10b)

So the new extended Jacobi elliptic function expansion method can obtain more new solutions, which contain solitary wave solutions, singular solitary solutions, and triangular function solutions. These new solutions cannot be obtained by other Jacobi elliptic function methods, such as Fu et al. [13], Fan [14], Yan [15–17], and Chen et al. [18–21].

3. Exact Solutions of the GSWW Equation

The Boussinesq approximation theory [25] for generalized classical shallow water wave leads to the GSWW equation: 𝑣_{𝑥𝑥𝑥𝑡}+𝛼𝑣_𝑥𝑣_𝑥𝑡+𝛽𝑣_𝑡𝑣_𝑥𝑥−𝑣_𝑥𝑡−𝑣_𝑥𝑥=0, (3.1) where 𝛼 and 𝛽 are nonzero constants. Using 𝑣_𝑥=𝑢 in [26], (3.1) can be simplified to 𝑢_𝑥𝑥𝑡+𝛼𝑢𝑢_𝑡−𝛽𝑢_𝑥𝜕_𝑥⁻¹𝑢_𝑡−𝑢_𝑡−𝑢_𝑥=0, (3.2) which has recently attracted many attentions from researchers under the following cases.

(1) For the case when 𝛼=2𝛽, the following AKNS-SWW equation: 𝑢_𝑥𝑥𝑡+2𝛽𝑢𝑢_𝑡−𝛽𝑢_𝑥𝜕_𝑥⁻¹𝑢_𝑡−𝑢_𝑡−𝑢_𝑥=0 (3.3) was discussed by Ablowitz et al. [27].

(2) For the case when 𝛼=𝛽, the following equation: 𝑢_𝑥𝑥𝑡+𝛽𝑢𝑢_𝑡−𝛽𝑢_𝑥𝜕_𝑥⁻¹𝑢_𝑡−𝑢_𝑡−𝑢_𝑥=0 (3.4) was discussed by Hirota and Satsuma [28].

The GSWW equation (3.1) in potential form was studied by Clarkson and Mansfield [26, 29] who gave a complete catalog of classical and nonclassical symmetry reductions. The necessary conditions of Painlevé tests were given by Weiss et al. [30] and the complete integrability of (3.1) for the case when 𝛼=2𝛽 and 𝛼=𝛽 was established by Ablowitz et al. [31]. It has been proven by Hietarinta [32] that the GSWW equation (3.2) can be expressed in Hirota’s bilinear form and when 𝛼=2𝛽, (3.2) can be reduced to (3.3) or to (3.4) when 𝛼=𝛽. In other words, both (3.3) and (3.4) are solvable by using Hirota’s bilinear method [33]. The 𝑁-soliton solutions for (3.3) and (3.4) had also been found using this technique [28]. Although there is no scaling transformation that can reduce (3.3) to (3.4), the classical methods of Lie, the nonclassical method of Bluman and Cole, and the direct method of Clarkson and Kruskal can be applied to solve the GSWW equation (3.2) to obtain some kinds of symmetry reductions [34, 35]. Recently, Elwakil found a lot of exact solutions by using the modified extended tanh-function method [36]. Yang and Hon employed a rational expansion to generalize Fan’s method for exact travelling wave solutions for the generalized shallow water wave (GSWW) equation [37].

According to the above method, to seek travelling wave solutions of (3.1), we make the following transformation: 𝑢(𝑥,𝑡)=𝑈(𝜉),𝜉=𝑘(𝑥−𝑐𝑡), (3.5) where 𝑘 and 𝑐 are constants to be determined later. Thus (3.1) becomes −𝑐𝑘²𝑈^′′′+𝑐𝛽−𝛼𝑈𝑈^′+(𝑐−1)𝑈^′=0. (3.6) Equation (3.6) is integrated at once yielding −𝑐𝑘²𝑈^′′+𝑐𝛽−𝛼2𝑈²+(𝑐−1)𝑈=0. (3.7) For the following, without loss of generality, we can set the integration constant equal to zero.

Now we consider the system (3.7) in the above three cases, so that (2.4a)–(2.4c). According to Step 2 and Step 3 in Section 2, by balancing the nonlinear term 𝑈² with the highest-order differential term 𝑈^′′ in (3.7), we suppose that (3.7) has the following formal solutions.

3.1. sn^′𝜉/sn𝜉 Expansion

Now we consider the ansätz (2.4a). For (3.7), the ansätz (2.4a) becomes 𝑢=𝑎₀+𝑎₁sn^′(𝜉)sn(𝜉)+𝑎₂sn^′(𝜉)sn(𝜉)²+𝑏₁sn^′(𝜉)sn(𝜉)⁻¹+𝑏₂sn^′(𝜉)sn(𝜉)⁻², (3.8) where 𝑎₀, 𝑎₁, 𝑎₂, 𝑏₁, 𝑏₂, 𝑘, 𝑙 and 𝑐 are constants to be determined later.

With the aid of Maple, substituting (3.8) along with (2.5) and (2.6) into (3.7), yields a set of algebraic equations for sn^𝑖𝜉cn^𝑗𝜉dn^𝑘𝜉(𝑖=0,1,2,…;𝑗=0,1;𝑘=0,1). Setting the coefficients of these terms sn^𝑖𝜉cn^𝑗𝜉dn^𝑘𝜉 to be zero yields a set of overdetermined algebraic equations with respect to 𝑎₀, 𝑎₁, 𝑎₂, 𝑏₁, 𝑏₂, 𝑘, 𝑙, and 𝑐.

By use of the Maple software package “Charsets” by Dongming Wang, which is based on the Wu-elimination method [24], solving the overdetermined algebraic equations, we get the following results.

Case 1. 𝑎₀=−1−48𝑐𝑘²−𝑐+2768𝑐²𝑘⁴+(𝑐−1)²𝑐𝛼−𝛽,𝑎₁=𝑎₂=𝑏₁𝑏=0,₂=364𝑐𝑘²−28𝑐𝑘²+768𝑐²𝑘⁴+(𝑐−1)²−(𝑐−1)²4𝑐²𝑘²,𝛼−𝛽𝑚=±𝑐−28𝑐𝑘²+768𝑐²𝑘⁴+(𝑐−1)².2𝑐𝑘 (3.9)

Case 2. 𝑎₀=−1−48𝑐𝑘²−𝑐−2768𝑐²𝑘⁴+(𝑐−1)²𝑐𝛼−𝛽,𝑎₁=𝑎₂=𝑏₁𝑏=0,₂=364𝑐𝑘²−28𝑐𝑘²−768𝑐²𝑘⁴+(𝑐−1)²−(𝑐−1)²4𝑐²𝑘²,𝛼−𝛽𝑚=±𝑐−28𝑐𝑘²−768𝑐²𝑘⁴+(𝑐−1)².2𝑐𝑘 (3.10)

Case 3. 𝑎₀=−1−48𝑐𝑘²−𝑐+2768𝑐²𝑘⁴+(𝑐−1)²𝑐𝛼−𝛽,𝑎₁=𝑏₁=𝑏₂=0,𝑎₂=−12𝑘²,𝛼−𝛽𝑚=±𝑐32𝑐𝑘²+768𝑐²𝑘⁴+(𝑐−1)².2𝑐𝑘 (3.11)

Case 4. 𝑎₀=−1−48𝑐𝑘²−𝑐−2768𝑐²𝑘⁴+(𝑐−1)²𝑐𝛼−𝛽,𝑎₁=𝑏₁=𝑏₂=0,𝑎₂=−12𝑘²,𝛼−𝛽𝑚=±𝑐32𝑐𝑘²−768𝑐²𝑘⁴+(𝑐−1)².2𝑐𝑘 (3.12)

Case 5. 𝑎₀2=−1+12𝑐𝑘²−𝑐+−192𝑐²𝑘⁴+(𝑐−1)²2𝑐𝛼−𝛽,𝑎₁=𝑏₁=0,𝑎₂=−12𝑘²,𝑏𝛼−𝛽₂=316𝑐𝑘²8𝑐𝑘²+−192𝑐²𝑘⁴+(𝑐−1)²−(𝑐−1)²64𝑐²𝑘²,𝛼−𝛽𝑚=±𝑐8𝑐𝑘²+−192𝑐²𝑘⁴+(𝑐−1)².4𝑐𝑘 (3.13)

Case 6. 𝑎₀2=−1+12𝑐𝑘²−𝑐−−192𝑐²𝑘⁴+(𝑐−1)²2𝑐𝛼−𝛽,𝑎₁=𝑏₁=0,𝑎₂=−12𝑘²,𝑏𝛼−𝛽₂=316𝑐𝑘²8𝑐𝑘²−−192𝑐²𝑘⁴+(𝑐−1)²−(𝑐−1)²64𝑐²𝑘²,𝛼−𝛽𝑚=±𝑐−28𝑐𝑘²−−192𝑐²𝑘⁴+(𝑐−1)².4𝑐𝑘 (3.14)
From (3.8) and Cases 1–6, we obtain the following solutions for (3.1).

Family 1. From (3.9), we obtain the following sn^′𝜉/sn𝜉 expansion solutions for the GSWW equation as follows: 𝑢₁=−1−48𝑐𝑘²−𝑐+2768𝑐²𝑘⁴+(𝑐−1)²𝑐+3𝛼−𝛽64𝑐𝑘²−28𝑐𝑘²+768𝑐²𝑘⁴+(𝑐−1)²−(𝑐−1)²4𝑐²𝑘²𝛼−𝛽sc²(𝜉)nd²(𝜉), (3.15) where 𝑚=±𝑐[−28𝑐𝑘²+768𝑐²𝑘⁴+(𝑐−1)²]/2𝑐𝑘 and 𝑐 and 𝑘 are arbitrary constants.

Family 2. From (3.10), we obtain the following sn^′𝜉/sn𝜉 expansion solutions for the GSWW equation as follows: 𝑢₂=−1−48𝑐𝑘²−𝑐−2768𝑐²𝑘⁴+(𝑐−1)²𝑐+3𝛼−𝛽64𝑐𝑘²−28𝑐𝑘²−768𝑐²𝑘⁴+(𝑐−1)²−(𝑐−1)²4𝑐²𝑘²𝛼−𝛽sc²(𝜉)nd²(𝜉), (3.16) where 𝑚=±𝑐[−28𝑐𝑘²−768𝑐²𝑘⁴+(𝑐−1)²]/2𝑐𝑘 and 𝑐 and 𝑘 are arbitrary constants.

Family 3. From (3.11), we obtain the following sn^′𝜉/sn𝜉 expansion solutions for the GSWW equation as follows: 𝑢₃=−1−48𝑐𝑘²−𝑐+2768𝑐²𝑘⁴+(𝑐−1)²𝑐−𝛼−𝛽12𝑘²𝛼−𝛽cs²(𝜉)dn²(𝜉), (3.17) where 𝑚=±𝑐[32𝑐𝑘²+768𝑐²𝑘⁴+(𝑐−1)²]/2𝑐𝑘 and 𝑐 and 𝑘 are arbitrary constants.

Family 4. From (3.12), we obtain the following sn^′𝜉/sn𝜉 expansion solutions for the GSWW equation as follows: 𝑢₄=−1−48𝑐𝑘²−𝑐−2768𝑐²𝑘⁴+(𝑐−1)²𝑐−𝛼−𝛽12𝑘²𝛼−𝛽cs²(𝜉)dn²(𝜉), (3.18) where 𝑚=±𝑐[32𝑐𝑘²−768𝑐²𝑘⁴+(𝑐−1)²]/2𝑐𝑘 and 𝑐 and 𝑘 are arbitrary constants.

Family 5. From (3.13), we obtain the following sn^′𝜉/sn𝜉 expansion solutions for the GSWW equation as follows: 𝑢₅2=−1+12𝑐𝑘²−𝑐+−192𝑐²𝑘⁴+(𝑐−1)²−2𝑐𝛼−𝛽12𝑘²𝛼−𝛽cs²(𝜉)dn²(𝜉)+316𝑐𝑘²8𝑐𝑘²+−192𝑐²𝑘⁴+(𝑐−1)²−(𝑐−1)²64𝑐²𝑘²𝛼−𝛽sc²(𝜉)nd²(𝜉), (3.19) where 𝑚=±𝑐[8𝑐𝑘²+−192𝑐²𝑘⁴+(𝑐−1)²]/4𝑐𝑘 and 𝑐 and 𝑘 are arbitrary constants.

Family 6. From (3.14), we obtain the following sn^′𝜉/sn𝜉 expansion solutions for the GSWW equation as follows: 𝑢₆2=−1+12𝑐𝑘²−𝑐−−192𝑐²𝑘⁴+(𝑐−1)²−2𝑐𝛼−𝛽12𝑘²𝛼−𝛽cs²(𝜉)dn²(𝜉)+316𝑐𝑘²8𝑐𝑘²−−192𝑐²𝑘⁴+(𝑐−1)²−(𝑐−1)²64𝑐²𝑘²𝛼−𝛽sc²(𝜉)nd²(𝜉), (3.20) where 𝑚=±𝑐[−28𝑐𝑘²−−192𝑐²𝑘⁴+(𝑐−1)²]/4𝑐𝑘 and 𝑐 and 𝑘 are arbitrary constants.

Remark 3.1. Here we find that the modulus 𝑚 of the Jacobi elliptic functions has relations with 𝑐 and 𝑘 in Cases 1–6. To further analyze their relations, we take the solutions (3.9), (3.11), and (3.13) as samples by three figures (see Figures 1, 2, and 3).

Figure 1 

Figure of 𝑚 in (3.9).

Figure 2 

Figure of 𝑚 in (3.11).

Figure 3 

Figure of 𝑚 in (3.13).

3.2. cn^′𝜉/cn𝜉  Expansion

Now we consider the ansätz (2.4b). For (3.7), the ansätz (2.4b) becomes 𝑢=𝑎₀+𝑎₁cn^′(𝜉)cn(𝜉)+𝑎₂cn^′(𝜉)cn(𝜉)²+𝑏₁cn^′(𝜉)cn(𝜉)⁻¹+𝑏₂cn^′(𝜉)cn(𝜉)⁻², (3.21) where 𝑎₀, 𝑎₁, 𝑎₂, 𝑏₁, 𝑏₂, 𝑘, 𝑙, and 𝑐 are constants to be determined later. Following the same steps in Section 3.1, we can obtain the following cn^′𝜉/cn𝜉 expansion solutions.

Family 7. 𝑢₇=𝑐−1+48𝑐²𝑘⁴+(𝑐−1)²𝑐−𝛼−𝛽12𝑘²𝛼−𝛽cs²(𝜉)nd²(𝜉), (3.22) where 𝑚=±𝑐(8𝑐𝑘²+48𝑐²𝑘⁴+(𝑐−1)²)/4𝑐𝑘 and 𝑐 and 𝑘 are arbitrary constants.

Family 8. 𝑢₈=𝑐−1−48𝑐²𝑘⁴+(𝑐−1)²𝑐−𝛼−𝛽12𝑘²𝛼−𝛽cs²(𝜉)nd²(𝜉), (3.23) where 𝑚=±𝑐(8𝑐𝑘²−48𝑐²𝑘⁴+(𝑐−1)²)/4𝑐𝑘 and 𝑐 and 𝑘 are arbitrary constants.

Family 9. 𝑢₉=𝑐−1+48𝑐²𝑘⁴+(𝑐−1)²𝑐−𝛼−𝛽12𝑘²𝛼−𝛽sc²(𝜉)dn²(𝜉), (3.24) where 𝑚=±𝑐(8𝑐𝑘²+48𝑐²𝑘⁴+(𝑐−1)²)/4𝑐𝑘 and 𝑐 and 𝑘 are arbitrary constants.

Family 10. 𝑢₁₀=𝑐−1−48𝑐²𝑘⁴+(𝑐−1)²𝑐−𝛼−𝛽12𝑘²𝛼−𝛽sc²(𝜉)dn²(𝜉), (3.25) where 𝑚=±𝑐(8𝑐𝑘²−48𝑐²𝑘⁴+(𝑐−1)²)/4𝑐𝑘 and 𝑐 and 𝑘 are arbitrary constants.

Family 11. 𝑢₁₁=𝑐−1+−192𝑐²𝑘⁴+(𝑐−1)²𝑐−𝛼−𝛽12𝑘²𝛼−𝛽sc²(𝜉)dn²(𝜉)+cs²(𝜉)nd²(𝜉), (3.26) where 𝑚=±𝑐(8𝑐𝑘²+−192𝑐²𝑘⁴+(𝑐−1)²)/4𝑐𝑘 and 𝑐 and 𝑘 are arbitrary constants.

Family 12. 𝑢₁₂=𝑐−1−−192𝑐²𝑘⁴+(𝑐−1)²𝑐−𝛼−𝛽12𝑘²𝛼−𝛽sc²(𝜉)dn²(𝜉)+cs²(𝜉)nd²(𝜉), (3.27) where 𝑚=±𝑐(8𝑐𝑘²−−192𝑐²𝑘⁴+(𝑐−1)²)/4𝑐𝑘 and 𝑐 and 𝑘 are arbitrary constants.

3.3. dn^′𝜉/dn𝜉 Expansion

Now we consider the ansätz (2.4c). For (3.7), the ansätz (2.4c) becomes 𝑢=𝑎₀+𝑎₁dn^′(𝜉)dn(𝜉)+𝑎₂dn^′(𝜉)dn(𝜉)²+𝑏₁dn^′(𝜉)dn(𝜉)⁻¹+𝑏₂dn^′(𝜉)dn(𝜉)⁻², (3.28) where 𝑎₀, 𝑎₁, 𝑏₁, 𝑎₂, 𝑏₂, 𝑘, 𝑙, and 𝑐 are constants to be determined later. Following the same steps in Section 3.1, we can obtain the following dn^′𝜉/dn𝜉 expansion solutions.

Family 13. 𝑢₁₃=3−1792𝑐²𝑘⁴−(𝑐−1)²−64𝑐𝑘²768𝑐²𝑘⁴+(𝑐−1)²4𝑚⁴𝑐²𝑘²𝛼−𝛽ds²(𝜉)nc²(𝜉)+𝑐−1−48𝑐𝑘²−2768𝑐²𝑘⁴+(𝑐−1)²𝑐,𝛼−𝛽 (3.29) where 𝑚=±𝑐(32𝑐𝑘²+768𝑐²𝑘⁴+(𝑐−1)²)/2𝑐𝑘 and 𝑐 and 𝑘 are arbitrary constants.

Family 14. 𝑢₁₄=3−1792𝑐²𝑘⁴−(𝑐−1)²+64𝑐𝑘²768𝑐²𝑘⁴+(𝑐−1)²4𝑚⁴𝑐²𝑘²𝛼−𝛽ds²(𝜉)nc²(𝜉)+𝑐−1−48𝑐𝑘²+2768𝑐²𝑘⁴+(𝑐−1)²𝑐,𝛼−𝛽 (3.30) where 𝑚=±𝑐(32𝑐𝑘²−768𝑐²𝑘⁴+(𝑐−1)²)/2𝑐𝑘 and 𝑐 and 𝑘 are arbitrary constants.

Family 15. 𝑢₁₅=𝑐−1−48𝑐𝑘²−2768𝑐²𝑘⁴+(𝑐−1)²𝑐−𝛼−𝛽12𝑚⁴𝑘²𝛼−𝛽sd²(𝜉)cn²(𝜉), (3.31) where 𝑚=±𝑐(32𝑐𝑘²+768𝑐²𝑘⁴+(𝑐−1)²)/2𝑐𝑘 and 𝑐 and 𝑘 are arbitrary constants.

Family 16. 𝑢₁₆=𝑐−1−48𝑐𝑘²+2768𝑐²𝑘⁴+(𝑐−1)²𝑐−𝛼−𝛽12𝑚⁴𝑘²𝛼−𝛽sd²(𝜉)cn²(𝜉), (3.32) where 𝑚=±𝑐(32𝑐𝑘²−768𝑐²𝑘⁴+(𝑐−1)²)/2𝑐𝑘 and 𝑐 and 𝑘 are arbitrary constants.

Family 17. 𝑢₁₇=2𝑐−2+24𝑐𝑘²−−192𝑐²𝑘⁴+(𝑐−1)²−2𝑐𝛼−𝛽12𝑚⁴𝑘²𝛼−𝛽sd²(𝜉)cn²(𝜉)+3−(𝑐−1)²+128𝑐²𝑘⁴−16𝑐𝑘²−192𝑐²𝑘⁴+(𝑐−1)²64𝑚⁴𝑐²𝑘²𝛼−𝛽ds²(𝜉)nc²(𝜉), (3.33) where 𝑚=±𝑐(8𝑐𝑘²+−192𝑐²𝑘⁴+(𝑐−1)²)/4𝑐𝑘 and 𝑐 and 𝑘 are arbitrary constants.

Family 18. 𝑢₁₈=2𝑐−2+24𝑐𝑘²+−192𝑐²𝑘⁴+(𝑐−1)²−2𝑐𝛼−𝛽12𝑚⁴𝑘²𝛼−𝛽sd²(𝜉)cn²(𝜉)+3−(𝑐−1)²+128𝑐²𝑘⁴+16𝑐𝑘²−192𝑐²𝑘⁴+(𝑐−1)²64𝑚⁴𝑐²𝑘²𝛼−𝛽ds²(𝜉)nc²(𝜉), (3.34) where 𝑚=±𝑐(8𝑐𝑘²−−192𝑐²𝑘⁴+(𝑐−1)²)/4𝑐𝑘 and 𝑐 and 𝑘 are arbitrary constants.

Remark 3.2. The solutions obtained here, to our knowledge, are all new families of periodic solution of the GSWW equation.

Remark 3.3. In order to understand the significance of these solutions in Families 1 and 18, here we take the solutions (3.15), (3.17), (3.19), (3.22), (3.24), (3.26), (3.29), (3.31), and (3.33) as samples to further analyze their properties by some figures (see Figures 4, 5, 6, 7,8, 9, 10, 11, and 12).

Figure 4 

The solution 𝑢1 of GSWW equation with 𝑘=0.3, 𝑐=2, 𝛼=3, 𝛽=1.

Figure 5 

The solution 𝑢3 of GSWW equation with 𝑘=0.3, 𝑐=2, 𝛼=3, 𝛽=1.

Figure 6 

The solution 𝑢5 of GSWW equation with 𝑘=0.1, 𝑐=2, 𝛼=3, 𝛽=1.

Figure 7 

The solution 𝑢7 of GSWW equation with 𝑘=0.3, 𝑐=2, 𝛼=3, 𝛽=1.

Figure 8 

The solution 𝑢9 of GSWW equation with 𝑘=0.3, 𝑐=2, 𝛼=3, 𝛽=1.

Figure 9 

The solution 𝑢11 of GSWW equation with 𝑘=0.1, 𝑐=2, 𝛼=3, 𝛽=1.

Figure 10 

The solution 𝑢13 of GSWW equation with 𝑘=0.3, 𝑐=2, 𝛼=3, 𝛽=1.

Figure 11 

The solution 𝑢15 of GSWW equation with 𝑘=0.3, 𝑐=2, 𝛼=3, 𝛽=1.

Figure 12 

The solution 𝑢17 of GSWW equation with 𝑘=0.1, 𝑐=2, 𝛼=3, 𝛽=1.

4. Conclusion and More General Form

In short, we have presented the new extended Jacobi elliptic function expansion method. The GSWW equation is chosen to illustrate the method such that many families of new Jacobi elliptic function solutions are obtained. When the modulus 𝑚→1 or 0, some of these obtained solutions degenerate as solitary solutions or trigonometric function solutions. The algorithm can be applied to many nonlinear differential equations in mathematical physics. Other types of solutions of (2.1) need to be studied further.

It is easy to see that above-mentioned method is only applied to these nonlinear ODEs with constant coefficients or nonlinear partial differential equations, which can be reduced to the corresponding nonlinear ODEs with constant coefficients by using some transformations; otherwise, the method will not work. In order to overcome the disadvantage of the method, we change the method into a general form as follows.

If we do not reduce (2.1) to a nonlinear ODE (2.3) with constant coefficients, then we directly assume that (2.1) has the following solutions:(1)sn^′(𝜉)/sn(𝜉) expansion:𝑢(𝑥,𝑡)=𝑎₀(𝑥,𝑡)+^𝑛_𝑖=1𝑎_𝑖(𝑥,𝑡)sn^′(𝜑(𝑥,𝑡))sn(𝜑(𝑥,𝑡))^𝑖+𝑏_𝑖(𝑥,𝑡)sn^′(𝜑(𝑥,𝑡))sn(𝜑(𝑥,𝑡))^−𝑖, (4.1a)(2)cn^′(𝜉)/cn(𝜉) expansion: 𝑢(𝑥,𝑡)=𝑎₀(𝑥,𝑡)+^𝑛_𝑖=1𝑎_𝑖(𝑥,𝑡)cn^′(𝜑(𝑥,𝑡))cn(𝜑(𝑥,𝑡))^𝑖+𝑏_𝑖(𝑥,𝑡)cn^′(𝜑(𝑥,𝑡))cn(𝜑(𝑥,𝑡))^−𝑖, (4.1b)(3)dn^′𝜉/dn𝜉 expansion: 𝑢(𝑥,𝑡)=𝑎₀(𝑥,𝑡)+^𝑛_𝑖=1𝑎_𝑖(𝑥,𝑡)dn^′(𝜑(𝑥,𝑡))dn(𝜑(𝑥,𝑡))^𝑖+𝑏_𝑖(𝑥,𝑡)dn^′(𝜑(𝑥,𝑡))dn(𝜑(𝑥,𝑡))^−𝑖, (4.1c)where 𝑎₀(𝑥,𝑡), 𝑎_𝑖(𝑥,𝑡), 𝑏_𝑖(𝑥,𝑡) (𝑖=1,2,…,𝑛), and 𝜑(𝑥,𝑡) are functions to be determined later. Substituting (4.1a), (4.1b), and (4.1c) with (2.5) and (2.6) into (2.1) yields a set of nonlinear partial differential equations with respect to these unknown functions 𝑎₀(𝑥,𝑡), 𝑎_𝑖(𝑥,𝑡), 𝑏_𝑖(𝑥,𝑡)(𝑖=1,2,…,𝑛), and 𝜑(𝑥,𝑡), respectively. If we can solve these functions from the obtained set of nonlinear partial differential equations, then we may obtain more types of doubly periodic solutions of (2.1). The more general form is better applied to nonlinear differential equations with variable coefficients. Particularly, if we only deduce the conclusions that all these function 𝑎₀(𝑥,𝑡), 𝑎_𝑖(𝑥,𝑡), 𝑏_𝑖(𝑥,𝑡) (𝑖=1,2,…) are all constants and 𝜑(𝑥,𝑡) is of the form 𝑘(𝑥−𝑐𝑡) (𝑘, 𝑐 constants), then the obtained results are the same as the ones found by using the method presented in Section 2. About the applications of the more general form, we will give some examples in future.

Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgments

Many thanks are due to suggestions from the referees and support from the National Natural Science Foundation of China (Grant no. 61171179) and the Young Scientists Fund of the National Natural Science Foundation of China (Grant no. 10901145).