New Exact Solutions for a Higher-Order Wave Equation of KdV Type Using Extended F-Expansion Method

Abstract

The F-expansion method is used to find traveling wave solutions to various wave equations. By giving more solutions of the general subequation, an extended F-expansion method is introduced by Emmanuel. In our work, a generalized KdV type equation of neglecting the highest-order infinitesimal term, which is an important water wave model, is discussed by using the extended F-expansion method. And when the parameters satisfy certain relations, some new exact solutions expressed by Jacobi elliptic function, hyperbolic function, and trigonometric function are obtained. The related results are enriched.

1. Introduction

It has recently become more interesting to obtain exact solutions of nonlinear partial differential equations. These equations are mathematical models of complex physical phenomena that arise in engineering, applied mathematics, chemistry, biology, mechanics, physics, and so forth. Thus, the investigation of the traveling wave solutions to nonlinear evolution equations (NLEEs) plays an important role in mathematical physics. A lot of physical models have supported a wide variety of solitary wave solutions.

In the recent years, much effort has been spent on this task and many significant methods have been established such as inverse scattering transform [1], Backlund and Darboux transform [2], Hirota [3], homogeneous balance method [4], Jacobi elliptic function method [5], tanh-function method [6], exp-function method [7], simple equation method [8], F-expansion method [9, 10], improved F-expansion method [11], and extended F-expansion method [12].

Wang and Li [10] developed a new algebraic method, belonging to the simplest equation method [13–16], to seek more new solutions of NLEEs that can be expressed as polynomial in an elementary function which satisfies a more general subequation than other subequations like Riccati equation, auxiliary ordinary equation, elliptic equation, and generalized Riccati equation. The Fans method not only gives a unified formation to construct various traveling wave solutions but also provides a guideline to classify the various types of traveling wave solutions according to five parameters. An extended F-expansion method was proposed by Yomba in 2005 by giving more solutions of the general subequation. Using the new method, exact solutions of many NLEEs are successfully obtained [12].

In this work, we apply the extended F-expansion method on a higher-order wave equation of KdV type for obtaining new exact traveling solutions.

In 1995, based on the physical and asymptotic considerations, Fokas [17] derived the following generalized KdV equation: 𝜂_𝑡+𝜂_𝑥+𝛼𝜂𝜂_𝑥+𝛽𝜂_𝑥𝑥𝑥+𝜌₁𝛼²𝜂²𝜂_𝑥+𝛼𝛽𝜌₂𝜂𝜂_𝑥𝑥𝑥+𝜌₃𝜂_𝑥𝜂_𝑥𝑥+𝜌₄𝛼³𝜂³𝜂_𝑥+𝛼²𝛽𝜌₅𝜂²𝜂_𝑥𝑥𝑥+𝜌₆𝜂𝜂_𝑥𝜂_𝑥𝑥+𝜌₇𝜂_𝑥³=0, (1) which is an important water wave model, where 𝛼=3𝐴/2,𝛽=𝐵/6,𝜌₁=−1/6,𝜌₆=5/3,𝜌₃ = 23/6,𝜌₄=1/8,𝜌₅=7/18,𝜌₆=79/36,  and 𝜌₇=45/36. Regarding the 𝜌₁,  𝜌₂,  𝜌₃,  𝜌₄,  𝜌₅,  𝜌₆,  and 𝜌₇ as free parameters and using the 𝜌₄ to replace the 𝜌₄𝛼², (1) becomes the following PDE: 𝑢_𝑡+𝑢_𝑥+𝛼𝑢𝑢_𝑥+𝛽𝑢_𝑥𝑥𝑥+𝜌₁𝛼²𝑢²𝑢_𝑥+𝛼𝛽𝜌₂𝑢𝑢_𝑥𝑥𝑥+𝜌₃𝑢_𝑥𝜂_𝑥𝑥+𝜌₄𝛼𝑢³𝑢_𝑥+𝛼²𝛽𝜌₅𝑢²𝑢_𝑥𝑥𝑥+𝜌₆𝑢𝑢_𝑥𝑢_𝑥𝑥+𝜌₇𝜂_𝑥³=0, (2) which is given by Tzirtzilakis et al. in [18]. They called it high-order wave equation of KdV type. Just as Tzirtzilakis et al. [18] said, these two equations are both water wave equations of KdV type, which are more physically and practically meaningful.

Assuming that the waves are unidirectional and neglecting terms of 𝑂(𝛼²,𝛽³,𝛼𝛽), (1) can be reduced to the classical KdV equation: 𝜂_𝑡+𝜂_𝑥+𝛼𝜂𝜂_𝑥+𝛽𝜂_𝑥𝑥𝑥=0. (3) In [17], Fokas assumed that 𝑂(𝛽) is less than 𝑂(𝛼). According to this assumption, we easily know that 𝑂(𝛼²𝛽)<𝑂(𝛼³) and 𝑂(𝛼²𝛽)<𝑂(𝛼𝛽). Neglecting two high-order infinitesimal terms of 𝑂(𝛼³,𝛼²𝛽), (1) can be reduced to another high-order wave equation of KdV type [18, 19] as follows: 𝜂_𝑡+𝜂_𝑥+𝛼𝜂𝜂_𝑥+𝛽𝜂_𝑥𝑥𝑥+𝜌₁𝛼²𝜂²𝜂_𝑥+𝛼𝛽𝜌₂𝜂𝜂_𝑥𝑥𝑥+𝜌₃𝜂_𝑥𝜂_𝑥𝑥=0. (4) Equation (4) is a special case of (1) for 𝜌₄=𝜌₅=𝜌₆=𝜌₇=0. Equation (4) was first derived in [20] by using the method of bi-Hamiltonian systems, whose Lax pair was also given in [21].

If only we neglect the highest-order infinitesimal term of 𝑂(𝛼²𝛽), then (1) can be reduced to a new generalized KdV equation as follows: 𝜂_𝑡+𝜂_𝑥+𝛼𝜂𝜂_𝑥+𝛽𝜂_𝑥𝑥𝑥+𝜌₁𝛼²𝜂²𝜂_𝑥+𝛼𝛽𝜌₂𝜂𝜂_𝑥𝑥x+𝜌₃𝜂_𝑥𝜂_𝑥𝑥+𝜌₄𝛼³𝜂³𝜂_𝑥=0. (5)

In fact, (5) is another special case of (1) for 𝜌₅=𝜌₆=𝜌₇=0. It is also third-order approximate equation of KdV type. Of course, on describing dynamical behaviors of water waves, (4) is only a rough approximative model of (1) compared with (5); that is, the precision of model (5) is better than that of model (4) on describing dynamical behaviors of water waves. In other words, the model (5) exhibits much richer phenomenology than the model (4). Therefore, the investigation of exact traveling wave solutions for (5) is more practically meaningful than that of (4).

Equation (1) is studied by many researchers and some useful results are obtained when 𝜌_𝑖 takes special values. However, by using the current methods, we cannot obtain exact solutions of (1) in universal conditions. Therefore, the investigation of exact solutions of (5) is necessary and important. Equation (5) is perhaps not integrable. But it would be interesting to check its asymptotic integrability [22]. Equation (5) is studied by Wu et al. in [23] using the integral bifurcation method and some exact solutions in parameter form are given. In this paper, regarding the 𝜌_𝑖(𝑖=1,2,3,4) as free parameters and by using the extended F-expansion method [12], we will investigate exact traveling wave solutions of (5).

The organization of the paper is as follows. In Section 2, a brief description of the extended F-expansion for finding traveling wave solutions of nonlinear equations is given. In Section 3, we will study (5) by the extended F-expansion methods. Finally conclusions are given in Section 4.

2. Description of the Extended F-Expansion Methods

Based on F-expansion method, the main procedures of the extended F-expansion method are as follows [12].

Step 1. Consider a general nonlinear PDE in the form 𝐹𝑢,𝑢_𝑥,𝑢_𝑡,𝑢_𝑥𝑥,𝑢_𝑥𝑡,…=0. (6) Using 𝑢(𝑥,𝑡)=𝑈(𝜉), 𝜉=𝑥−𝑐𝑡, we can rewrite (6) as the following nonlinear ODE: 𝐹𝑈,𝑈^′,𝑈^′′,⋯=0, (7) where the prime denotes differentiation with respect to 𝜉.

Step 2. Suppose that the solution of ODE (7) can be written as follows: 𝑈(𝜉)=𝐴₀+^𝑛_𝑖=1𝐴_𝑖𝐹^𝑖(𝜉)+𝐵_𝑖𝐹^−𝑖(𝜉), (8) where 𝐴_𝑖, 𝐵_𝑖(𝑖=1,2,…𝑛) are constants to be determined later, 𝑛 is a positive integer that is given by the homogeneous balance principle, and 𝐹(𝜉) satisfies the following equation: 𝐹^′(𝜉)²=ℎ₀+ℎ₁𝐹(𝜉)+ℎ₂𝐹²(𝜉)+ℎ₃𝐹³(𝜉)+ℎ₄𝐹⁴(𝜉), (9) where ℎ₀, ℎ₁, ℎ₂, ℎ₃, and ℎ₄ are constant.

Step 3. Substituting (8) along with (9) into (7) and then setting all the coefficients of 𝐹^𝑗(𝜉)(𝑗=0,1,2,…,) of the resulting system’s numerator to zero yield a set of overdetermined nonlinear algebraic equations for 𝐴₀, 𝐴_𝑖, and 𝐵_𝑖(𝑖=1,2,…,𝑛).

Step 4. Assuming that the constants 𝐴₀, 𝐴_𝑖, and 𝐵_𝑖(𝑖=1,2,…𝑛) can be obtained by solving the algebraic equations in Step 3 and then substituting these constants and the solutions of (9), depending on the special conditions chosen for the ℎ₀, ℎ₁, ℎ₂, ℎ₃, and ℎ₄, we can obtain the explicit solutions of (6) immediately.

3. Exact Solutions of (5)

Making a transformation 𝜂(𝑡,𝑥)=𝜙(𝜉) with 𝜉=𝑥−𝑐𝑡, (5) can be reduced to the following ODE: −𝑐𝜙^′+𝜙^′+𝛼𝜙𝜙^′+𝛽𝜙^′′′+𝜌₁𝛼²𝜙²𝜙^′+𝛼𝛽𝜌₂𝜙𝜙^′′′+𝜌₃𝜙^′𝜙^′′+𝜌₄𝛼³𝜙³𝜙^′=0, (10) where 𝑐 is wave velocity which moves along the direction of 𝑥-axis and 𝑐≠0. Integrating (10) once and setting the integral constant as 𝑅 yield ()11−𝑐𝜙+2𝛼𝜙²+𝛽𝜙^′′+13𝜌₁𝛼²𝜙³+𝛼𝛽𝜌₂𝜙𝜙^′′+12𝜌₃−𝜌₂𝜙^′2+14𝜌₄𝛼³𝜙⁴+𝑅=0. (11)

Suppose that (11) owns the solutions in the form 𝜙(𝜉)=𝐴₀+𝐴₁𝐹(𝜉)+𝐵₁𝐹(𝜉). (12)

Substituting (12) and (9) into (11) and then setting all the coefficients of 𝐹^𝑘(𝑘=−5,…,5) of the resulting system to zero, we can obtain the following results.

3.1. Case of  ℎ₁=ℎ₃=0

In this situation, we have the following cases.

Case 1. Consider 𝐴₀=−2𝛽ℎ₂𝛼,𝐴₁=±2𝛽−6ℎ₂ℎ₄𝛼,𝐵₁=0,𝜌₁=𝜌₃2+𝜌₂+14𝛽ℎ₂,𝜌₄=𝜌₃+3𝜌₂12𝛽ℎ₂4,𝑐=3𝛽²ℎ²₂𝜌₃+1, (13) where ℎ₂≠0 and ℎ₀ and ℎ₄ are arbitrary constants.

Case 2. Consider 𝐴₀=−2𝛽ℎ₂𝛼,𝐴₁𝐵=0,₁=±2𝛽−6ℎ₀ℎ₂𝛼,𝜌₁=𝜌₃2+𝜌₂+14𝛽ℎ₂,𝜌₄=𝜌₃+3𝜌₂12𝛽ℎ₂4,𝑐=3𝛽²ℎ²₂𝜌₃+1, (14) where ℎ₂≠0 and ℎ₀ and ℎ₄ are arbitrary constants.

Case 3. Also 𝐴₀=𝛼𝐵²₁12𝛽ℎ₀,𝐴₁=𝑤𝐵₁,𝜌₁=−6𝛽ℎ₀𝛼²𝐵²₁−12𝜌₃,𝜌₂=−𝜌₃,𝜌₄=𝑚4𝛽𝜌₃ℎ₀𝛼²𝐵²₁,𝜌𝑐=−³𝐵⁴₁𝛼⁴864𝛽²ℎ²₀+𝛼²𝐵²₁24𝛽ℎ₀+1−6𝛽ℎ₀1𝑤+2𝑤𝐵²₁𝜌₃𝛼²+𝛽ℎ₂−𝐵²₁𝜌₃𝛼²ℎ₂12ℎ₀, (15) where 𝑤=±ℎ₄/ℎ₀, and ℎ₀≠0, ℎ₂ and ℎ₄ are arbitrary constants.

Substituting (13)–(15) into (12), we obtain, respectively, the following formal solution of (5): 𝜂(𝑥,𝑡)=2𝛽𝛼±−6ℎ₂ℎ₄𝐹(𝜉)−ℎ₂, (16) where 𝜉=𝑥−((4/3)𝛽²ℎ²₂𝜌₃+1)𝑡, 𝜌₁=𝜌₃/2+𝜌₂+1/4𝛽ℎ₂, and 𝜌₄=(𝜌₃+3𝜌₂)/12𝛽ℎ₂.

Moreover, 𝜂(𝑥,𝑡)=2𝛽𝛼±−6ℎ₀ℎ₂𝐹(𝜉)−ℎ₂, (17) where 𝜉=𝑥−((4/3)𝛽²ℎ²₂𝜌₃+1)𝑡, 𝜌₁=𝜌₃/2+𝜌₂+1/4𝛽ℎ₂, and 𝜌₄=(𝜌₃+3𝜌₂)/12𝛽ℎ₂.

We have 𝜂(𝑥,𝑡)=𝛼𝐵²₁12𝛽ℎ₀±ℎ₄ℎ₀𝐵₁𝐹(𝜉)+𝐵₁𝐹(𝜉), (18) where 𝜉=𝑥−𝑐𝑡, 𝑐 is determined in Case 3, 𝜌₁=−(6𝛽ℎ₀/𝛼²𝐵₁²)−(1/2)𝜌₃, 𝜌₂=−𝜌₃, and 𝜌₄=4𝛽𝜌₃ℎ₀/𝛼²𝐵₁².

When ℎ₁=ℎ₃=0, the general elliptic equation (9) is reduced to the auxiliary ordinary equation: 𝐹^′(𝜉)²=ℎ₀+ℎ₂𝐹²(𝜉)+ℎ₄𝐹⁴(𝜉). (19)

The solutions of (19) are given in Table 1. Combining (16)–(18) with Table 1, many exact solutions of (5) can be obtained. For simplicity, we just give out one case in Table 1, and the other cases can be discussed similarly.

Table 1 

Solutions of 𝐹(𝜉) in 𝐹^′2=ℎ₀+ℎ₂𝐹²+ℎ₄𝐹⁴.

When ℎ₀=1, ℎ₂=−(𝑚²+1), and ℎ₄=𝑚², the solution of (16) is 𝐹(𝜉)=sn(𝜉,𝑚) or 𝐹(𝜉)=cd(𝜉,𝑚). Substituting them into (17)–(19), we can obtain the following solutions of (5).

From (16), one has 𝜂(𝑥,𝑡)=2𝛽𝛼𝑚²+1±𝑚6𝑚²(),𝜂()=+1sn𝜉,𝑚𝑥,𝑡2𝛽𝛼𝑚²+1±𝑚6𝑚²+1cd(𝜉,𝑚), (20)(21) where 𝜉=𝑥−((4/3)𝛽²𝜌₃(𝑚²+1)²+1)𝑡, 𝜌₁=𝜌₃/2+𝜌₂−(1/4𝛽(𝑚²+1)), and 𝜌₄=−(𝜌₃+3𝜌₂)/12𝛽(𝑚²+1).

When 𝑚→1, sn(𝜉,𝑚)→tanh(𝜉), (20) becomes 𝜂(𝑥,𝑡)=4𝛽𝛼1±√3tanh(𝜉), (22) where 𝜉=𝑥−((16/3)𝛽²𝜌₃+1)𝑡, 𝜌₁=𝜌₃/2+𝜌₂−1/8𝛽, and 𝜌₄=−(𝜌₃+3𝜌₂)/24𝛽.

From (17), we have 𝜂(𝑥,𝑡)=2𝛽𝛼𝑚²+1±6𝑚²(),𝜂()=+1ns𝜉,𝑚𝑥,𝑡2𝛽𝛼𝑚²+1±6𝑚²+1dc(𝜉,𝑚), (23)(24) where 𝜉=𝑥−((4/3)𝛽²𝜌₃(𝑚²+1)²+1)𝑡, 𝜌₁=𝜌₃/2+𝜌₂−(1/4𝛽(𝑚²+1)), 𝜌₄=−(𝜌₃+3𝜌₂)/12𝛽(𝑚²+1).

When 𝑚→1, ns(𝜉,𝑚)→coth(𝜉), (23) becomes 𝜂(𝑥,𝑡)=4𝛽𝛼1±√3coth(𝜉), (25) where 𝜉=𝑥−((16/3)𝛽²𝜌₃+1)𝑡, 𝜌₁=𝜌₃/2+𝜌₂−1/8𝛽, and 𝜌₄=−(𝜌₃+3𝜌₂)/24𝛽.

When 𝑚→0, ns(𝜉,𝑚)→csc(𝜉), and dc(𝜉,𝑚)→sec(𝜉), (23) and (24) become 𝜂(𝑥,𝑡)=2𝛽𝛼1±√(𝜉),𝜂()=6csc𝑥,𝑡2𝛽𝛼1±√6sec(𝜉), (26) where 𝜉=𝑥−((4/3)𝛽²𝜌₃+1)𝑡, 𝜌₁=𝜌₃/2+𝜌₂−1/4𝛽, and 𝜌₄=−(𝜌₃+3𝜌₂)/12𝛽.

From (18), we have 𝜂(𝑥,𝑡)=𝛼𝐵²₁12𝛽±𝑚𝐵₁()+𝐵sn𝜉,𝑚¹(),𝜂()=sn𝜉,𝑚𝑥,𝑡𝛼𝐵²₁12𝛽±𝑚𝐵₁()+𝐵cd𝜉,𝑚¹(),cd𝜉,𝑚 (27)(28) where 𝜉=𝑥−𝑐𝑡, 𝑐 is determined in Case 3, 𝜌₁=−(6𝛽/𝛼²𝐵²₁)−(1/2)𝜌₃, 𝜌₂=−𝜌₃, and 𝜌₄=4𝛽𝜌₃/𝛼²𝐵²₁.

When 𝑚→1, sn(𝜉,𝑚)→tanh(𝜉), (27) becomes 𝜂(𝑥,𝑡)=𝛼𝐵²₁12𝛽±𝐵₁tanh(𝜉)+𝐵₁coth(𝜉), (29) where 𝜉=𝑥−𝑐𝑡, 𝜌₁=−(6𝛽/𝛼²𝐵²₁)−(1/2)𝜌₃, 𝜌₂=−𝜌₃, and 𝜌₄=4𝛽𝜌₃/𝛼²𝐵²₁.

When 𝑚→0, sn(𝜉,𝑚)→sin(𝜉), and dc(𝜉,𝑚)→sec(𝜉), (27) and (28) become 𝜂(𝑥,𝑡)=𝛼𝐵²₁12𝛽±𝐵₁(𝜉),𝜂()=csc𝑥,𝑡𝛼𝐵²₁12𝛽±𝐵₁sec(𝜉), (30) where 𝜉=𝑥−𝑐𝑡, 𝜌₁=−6𝛽/𝛼²𝐵²₁−(1/2)𝜌₃, 𝜌₂=−𝜌₃, and 𝜌₄=4𝛽𝜌₃/𝛼²𝐵₁².

3.2. Case of  ℎ₀=ℎ₁=0

In this situation, we have 𝐴₀=𝛽ℎ₃√𝑤−8ℎ₄ℎ₂+3ℎ²₃4𝛼ℎ₄,𝐴₁=𝛽√𝑤𝛼,𝐵₁𝜌=0,₁=𝜌₂+𝜌₃2−6ℎ₄𝛽𝑤,𝜌₄=−2ℎ₄3𝜌₂+𝜌₃,4𝛽𝑤𝑐=1+3𝜌₃𝛽²ℎ²₂−𝜌₃𝛽²ℎ₂ℎ₃4ℎ₃+√𝑤4ℎ₄+𝜌₃𝛽²ℎ³₃3ℎ₃+√𝑤16ℎ²₄, (31) where 𝑤=9ℎ²₃−24ℎ₂ℎ₄ and ℎ₂, ℎ₃, ℎ₄,𝜌₂, and 𝜌₃ are arbitrary constants.

Substituting (31) into (12), we obtain the following formal solution of (5): 𝜂()=𝛽ℎ𝑥,𝑡³√𝑤−8ℎ₄ℎ₂+3ℎ²₃4𝛼ℎ₄+√𝑤𝛽𝛼𝐹(𝜉), (32) where 𝜉=𝑥−𝑐𝑡 and 𝑐, 𝜌₁, and 𝜌₄ are determined in (31).

When ℎ₀=ℎ₁=0, the general elliptic equation (9) is reduced to the auxiliary ordinary equation: 𝐹^′(𝜉)²=ℎ₂𝐹²(𝜉)+ℎ₃𝐹³(𝜉)+ℎ₄𝐹⁴(𝜉). (33)

The solutions of (33) are given in Table 2.

Table 2 

Solutions of (33) with Δ=ℎ²₃−4ℎ₂ℎ₄, 𝜀=±1.

Combining (32) with Table 2, many exact solutions of (5) can be obtained. The following is an example.

If ℎ₂>0, solution of (33) is 𝐹(𝜉)=−ℎ₂ℎ₃sech²ℎ₂/2𝜉ℎ²₃−ℎ₂ℎ₄1+𝜀tanhℎ₂/2𝜉. (34) Therefore, solution of (4) is 𝜂()=𝛽ℎ𝑥,𝑡³√𝑤−8ℎ₂ℎ₄+3ℎ²₃4𝛼ℎ₄+𝛽√𝑤𝛼−ℎ₂ℎ₃sech²ℎ₂/2𝜉ℎ²₃−ℎ₂ℎ₄1+𝜀tanhℎ₂𝜉,/2 (35) where 𝜉=𝑥−𝑐𝑡, 𝑤=9ℎ²₃−24ℎ₂ℎ₄, and 𝑐, 𝜌₁, and 𝜌₄ are determined by (31).

3.3. Case of  ℎ₀≠0, ℎ₁≠0, ℎ₂≠0, ℎ₃≠0, and ℎ₄≠0

In this case, there exist three parameters 𝑟, 𝑝, and 𝑞 such that 𝐹^′(𝜉)²=ℎ₀+ℎ₁𝐹(𝜉)+ℎ₂𝐹²(𝜉)+ℎ₃𝐹³(𝜉)+ℎ₄𝐹⁴(𝜉)=𝑟+𝑝𝐹(𝜉)+𝑞𝐹²(𝜉)². (36) Equation (36) is satisfied only if the following relations hold: ℎ₀=𝑟²,ℎ₁ℎ=2𝑟𝑝,₂=2𝑟𝑞+𝑝²,ℎ₃=2𝑝𝑞,ℎ₄=𝑞². (37)

Equation (36) is the general Riccati equation. The solutions of (36) are listed in [12]. There are 24 group solutions named 𝜙^𝐼_𝑖(𝑖=1,2,…24), which we do not list for simplicity.

Substituting (36) and (9) into (11) and then setting all the coefficients of 𝐹^𝑘(𝑘=−4,…,4) of the resulting system to zero, we can obtain the following results.

Case 1. Consider 𝐴₀=𝛽𝑤𝑝+2𝑝²−8𝑞𝑟2𝛼,𝐴₁𝐵=0,₁=𝑤𝑟𝛽𝛼1,𝑐=1+3𝛽²𝜌₃𝑝²−4𝑞𝑟²,𝜌₁=𝜌₃2+𝜌₂−12𝛽𝑝²,𝜌−4𝑞𝑟₂=𝜌₂,𝜌₃=𝜌₃,𝜌₄𝜌=−³+3𝜌₂6𝛽𝑝²,−4𝑞𝑟 (38) where 𝑤=±2√3𝑝²−4𝑞𝑟 and 𝑝²−4𝑞𝑟>0.

Case 2. Consider 𝐴₀=𝛽𝑤𝑝+2𝑝²−8𝑞𝑟2𝛼,𝐴₁=𝑤𝑟𝛽𝛼,𝐵₁1=0,𝑐=1+3𝛽²𝜌₃𝑝²−4𝑞𝑟²,𝜌₁=𝜌₃2+𝜌₂−12𝛽𝑝²,𝜌−4𝑞𝑟₂=𝜌₂,𝜌₃=𝜌₃,𝜌₄𝜌=−³+3𝜌₂6𝛽𝑝²,−4𝑞𝑟 (39) where 𝑤=±2√3𝑝²−4𝑞𝑟 and 𝑝²−4𝑞𝑟>0.

Case 3. Consider 𝐴₀=𝛽𝛿𝑝+16𝑞𝑟+2𝑝²2𝛼,𝐴₁=𝛿𝛽𝑞𝛼,𝐵₁=𝑟𝛿𝛽𝛼1,𝑐=1−3𝜌₃𝛽²6𝑟𝑝𝛿𝑞−8𝑞𝑟+𝑝²²,𝜌₁=𝜌₃2+𝜌₂−12𝛽8𝑞𝑟+𝑝²,𝜌₂=𝜌₂,𝜌₃=𝜌₃,𝜌₄𝜌=−³+3𝜌₂6𝛽8𝑞𝑟+𝑝², (40) where 𝛿=±2√3𝑝²+8𝑞𝑟 and 𝑝²+8𝑞𝑟>0.

Substituting (38)–(40) into (12), we obtain, respectively, the following formal solution of (5): 𝜂()=𝛽𝑥,𝑡𝑤𝑝+2𝑝²−8𝑞𝑟+2𝛼𝑤𝑟𝛽𝛼𝐹(𝜉), (41) where 𝜉=𝑥−(1+(1/3)𝛽²𝜌₃(𝑝²−4𝑞𝑟)²)𝑡, 𝜌₁=(𝜌₃/2)+𝜌₂−1/2𝛽(𝑝²−4𝑞𝑟), and 𝜌₄=−(𝜌₃+3𝜌₂)/6𝛽(𝑝²−4𝑞𝑟).

Also 𝜂()=𝛽𝑥,𝑡𝑤𝑝+2𝑝²−8𝑞𝑟+2𝛼𝑤𝑟𝛽𝛼1𝐹(𝜉), (42) where 𝜉=𝑥−(1+(1/3)𝛽²𝜌₃(𝑝²−4𝑞𝑟)²)𝑡, 𝜌₁=(𝜌₃/2)+𝜌₂−1/2𝛽(𝑝²−4𝑞𝑟), and 𝜌₄=−(𝜌₃+3𝜌₂)/6𝛽(𝑝²−4𝑞𝑟).

Moreover, 𝜂()=𝛽𝑥,𝑡𝛿𝑝+16𝑞𝑟+2𝑝²+2𝛼𝛿𝛽𝑞𝛼𝐹(𝜉)+𝑟𝛿𝛽𝛼1𝐹(𝜉), (43) where 𝜉=𝑥−(1−(1/3)𝜌₃𝛽²(6𝑟𝑝𝛿𝑞−(8𝑞𝑟+𝑝²)²))𝑡, 𝜌₁=(𝜌₃/2)+𝜌₂−1/2𝛽(8𝑞𝑟+𝑝²), and 𝜌₄=−(𝜌₃+3𝜌₂)/6𝛽(8𝑞𝑟+𝑝²).

Substituting solutions of (36) 𝜙^𝐼_𝑖(𝑖=1,2,…,24) into (41)–(43), we can obtain a lot of solutions of (5). We just give one example.

When 𝑝²−4𝑝𝑞>0 and 𝑝𝑞≠0, 𝜙^𝐼₁=−(1/2𝑞)(𝑝+𝑝²−4𝑞𝑟tanh((𝑝²−4𝑞𝑟/2)𝜉)). Substituting 𝜙^𝐼₁ into (41) and (42), we have 𝜂()=𝛽𝑥,𝑡𝑤𝑝+2𝑝²−8𝑞𝑟+2𝛼𝑤𝑟𝛽𝛼𝜙^𝐼₁,𝜂()=𝛽𝑥,𝑡𝑤𝑝+2𝑝²−8𝑞𝑟+2𝛼𝑤𝑟𝛽𝛼1𝜙^𝐼₁, (44) where 𝜉=𝑥−𝑐𝑡 and 𝑐, 𝑤, 𝜌₁, and 𝜌₄ are determined by (38).

3.4. Case of  ℎ₀≠0, ℎ₁≠0, ℎ₂=0, ℎ₃≠0, and ℎ₄≠0

In this case, there exist three parameters 𝑟, 𝑝, and 𝑞 such that 𝐹^′(𝜉)²=ℎ₀+ℎ₁𝐹(𝜉)+ℎ₃𝐹³(𝜉)+ℎ₄𝐹⁴(𝜉)=𝑟+𝑝𝐹(𝜉)+𝑞𝐹²(𝜉)². (45) Equation (45) is satisfied only if the following relations hold: ℎ₀=𝑟²,ℎ₁ℎ=2𝑟𝑝,₃=2𝑝𝑞,ℎ₄=𝑞². (46) The following constraint should exist between 𝑟, 𝑝, and q parameters: 𝑝²=−2𝑞𝑟,𝑞𝑟<0. (47) Therefore, we can discuss the solution of (5) similarly as in the case shown in Section 3.3 under condition (47). Here, we omit it.

4. Conclusions

The investigation of the exact solutions of (5) is very important. And (5) is just studied by Wu et al. [23] using the integral bifurcation method. In our work, (5) is studied by extended F-expansion method and some new exact solutions expressed by Jacobi elliptic function, hyperbolic function, and trigonometric function are obtained. We believe that the results we obtained are useful in describing related physical phenomena. The correctness of all the solutions is verified by substituting them into original equation (5). Comparing with [23], it is easy to see that our method is more straightforward and the forms of the solutions obtained in our paper are also more simple. The related results are enriched.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by the Natural Science Foundation of China (no. 11161020 and no. 11361023) and the Natural Science Foundation of Yunnan (2013FZ117).