Exact Solutions of the Kudryashov-Sinelshchikov Equation Using the Multiple 𝐺^′/𝐺-Expansion Method

Abstract

Exact traveling wave solutions of the Kudryashov-Sinelshchikov equation are studied by the 𝐺^′/𝐺-expansion method and its variants. The solutions obtained include the form of Jacobi elliptic functions, hyperbolic functions, and trigonometric and rational functions. Many new exact traveling wave solutions can easily be derived from the general results under certain conditions. These methods are effective, simple, and many types of solutions can be obtained at the same time.

1. Introduction

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. Here, we study the Kudryashov-Sinelshchikov equation. In 2010, Kudryashov and Sinelshchikov [1] obtained a more common nonlinear partial differential equation for describing the pressure waves in a mixture liquid and gas bubbles taking into consideration the viscosity of liquid and the heat transfer, that is, 𝑢_𝑡+𝛼𝑢𝑢_𝑥+𝑢_𝑥𝑥𝑥−𝑢𝑢_𝑥𝑥𝑥−𝛽𝑢_𝑥𝑢_𝑥𝑥=0, (1) where 𝛼, 𝛽 are real parameters. In [2], they derived partial cases of nonlinear evolution equations of the fourth order for describing nonlinear pressure waves in a mixture liquid and gas bubbles. Some exact solutions are found and properties of nonlinear waves in a liquid with gas bubbles are discussed. Equation (1) is called Kudryashov-Sinelshchikov equation; it is generalization of the KdV and the BKdV equations and similar but not identical to the Camassa-Holm (CH) equation; it has been studied by some authors [1, 3–5]. Undistorted waves are governed by a corresponding ordinary differential equation which, for special values of some integration constant, is solved analytically in [1]. Solutions are derived in a more straightforward manner and cast into a simpler form, and some new types of solutions which contain solitary wave and periodic wave solutions are presented in [4]. Ryabov [5] obtained some exact solutions for 𝛽=−3 and 𝛽=−4 using a modification of the truncated expansion method [6, 7]. Li and He discussed the equation by the bifurcation method of dynamical systems and the method of phase portraits analysis [8–10]. In [11], the equation is studied by the Lie symmetry method.

The 𝐺^′/𝐺-expansion method proposed by Wang et al. [12] is one of the most effective direct methods to obtain travelling wave solutions of a large number of nonlinear evolution equations, such as the KdV equation, the mKdV equation, the variant Boussinesq equations, and the Hirota-Satsuma equations. Later, the further developed methods named the generalized 𝐺^′/𝐺-expansion method, the modified 𝐺^′/𝐺-expansion method, the extended 𝐺^′/𝐺-expansion method, and the improved 𝐺^′/𝐺-expansion method have been proposed in [13–15], respectively. The aim of this paper is to derive more and new traveling wave solutions of the Kudryashov-Sinelshchikov equation by the 𝐺^′/𝐺-expansion method and its variants.

The organization of the paper is as follows: in Section 2, a brief account of the 𝐺^′/𝐺-expansion and its variants that is, the generalized, improved, and extended versions, for finding the traveling wave solutions of nonlinear equations, is given. In Section 3, we will study the Kudryashov-Sinelshchikov equation by these methods. Finally conclusions are given in Section 4.

2. Description of Methods

2.1. The 𝐺^′/𝐺-Expansion Method

Step 1. Consider a general nonlinear PDE in the form 𝐹𝑢,𝑢_𝑥,𝑢_𝑡,𝑢_𝑥𝑥,𝑢_𝑥𝑡,…=0. (2) Using 𝑢(𝑥,𝑡)=𝑈(𝜉), 𝜉=𝑥−𝑐𝑡, we can rewrite (2) as the following nonlinear ODE: 𝐹𝑈,𝑈^′,𝑈^′′,…=0, (3) where the prime denotes differentiation with respect to 𝜉.

Step 2. Suppose that the solution of ODE (3) can be written as follows: 𝑈(𝜉)=^𝑛_𝑖=−𝑛𝑎_𝑖𝐺^′𝐺^𝑖, (4) where 𝜎, 𝑎_𝑖(𝑖=−𝑛,−𝑛+1,…) are constants to be determined later, 𝑛 is a positive integer, and 𝐺=𝐺(𝜉) satisfies the following second-order linear ordinary differential equation: 𝐺^′′+𝜆𝐺^′+𝜇𝐺=0, (5) where 𝜆, 𝜇 are real constants. The general solutions of (5) can be listed as follows.

When Δ=𝜆²−4𝜇>0, we obtain the hyperbolic function solution of (5) 𝐺(𝜉)=𝑒^{−(𝜆/2)𝜉}𝐴₁√coshΔ2𝜉+𝐴₂√sinhΔ2𝜉. (6)

When Δ=𝜆²−4𝜇<0, we obtain the trigonometric function solution of (5) 𝐺(𝜉)=𝑒^{−(𝜆/2)𝜉}𝐴₁√cos−Δ2𝜉+𝐴₂√sin−Δ2𝜉. (7)

When Δ=𝜆²−4𝜇=0, we obtain the solution of (5) 𝐺(𝜉)=𝑒^{−(𝜆/2)𝜉}𝐴₁+𝐴₂𝜉, (8) where 𝐴₁ and 𝐴₂ are arbitrary constants.

Step 3. Determine the positive integer 𝑛 by balancing the highest order derivatives and nonlinear terms in (3).

Step 4. Substituting (4) along with (5) into (3) and then setting all the coefficients of (𝐺^′/𝐺)^𝑘(𝑘=1,2,…) of the resulting system's numerator to zero yields a set of over-determined nonlinear algebraic equations for 𝑐 and 𝑎_𝑖(𝑖=−𝑛,−𝑛+1,…).

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

2.2. The Generalized 𝐺^′/𝐺-Expansion Method

In generalized version [13], one makes an ansatz for the solution 𝑈(𝜉) as 𝑈(𝜉)=^𝑛_𝑖=0𝑎_𝑖𝐺^′𝐺^𝑖, (9) where 𝐺=𝐺(𝜉) satisfies the following Jacobi elliptic equation: 𝐺^′2=𝑞𝐺⁴+𝑝𝐺²+𝑟, (10) where 𝑞, 𝑝, and 𝑟 are the arbitrary constants to be determined later and 𝑎_𝑛≠0. Substituting (9) into (3) and using (10), we obtain a polynomial in 𝐺^𝑖, 𝐺^′𝐺^𝑖(𝑖=1,2,…). Equating each coefficient of the resulted polynomials to zero yields a set of algebraic equations for 𝑎_𝑖, 𝑞, 𝑝, and 𝑟. Now, substituting 𝑎_𝑖 and the general solutions of (10) (see Table 1) into (9), we obtain many new traveling wave solutions in terms of Jacobi elliptic functions of the nonlinear PDE (2).

Table 1 

Relations between values of 𝑟, 𝑝, 𝑞 and corresponding 𝐺(𝜉) in (10).

2.3. The Extended 𝐺^′/𝐺-Expansion Method

In the extended form of this method [15], the solution 𝑈(𝜉) of (3) can be expressed as 𝑈(𝜉)=𝑎₀+^𝑛_𝑖=1𝑎_𝑖𝐺^′𝐺^𝑖+𝑏_𝑖𝐺^′𝐺^𝑖−1𝜎11+𝜇𝐺^′𝐺², (11) where 𝑎₀, 𝑎_𝑖, 𝑏_𝑖(𝑖=1,2,…,𝑛) are constants to be determined later, 𝜎=±1, 𝑛 is a positive integer, and 𝐺=𝐺(𝜉) satisfies the following second order linear ODE: 𝐺^′′+𝜇𝐺=0, (12) where 𝜇 is a constant. Substituting (11) into (3), using (12), collecting all terms with the same order of (𝐺^′/𝐺)^𝑘 and (𝐺^′/𝐺)^𝑘𝜎(1+(1/𝜇)(𝐺^′/𝐺)²) together, and then equating each coefficient of the resulting polynomial to zero yield a set of algebraic equations for 𝜇, 𝑎₀, 𝑎_𝑖, 𝑏_𝑖(𝑖=1,…,𝑛). On solving these algebraic equations, we obtain the values of the constants 𝜇, 𝑎₀, 𝑎_𝑖, 𝑏_𝑖(𝑖=1,…,𝑛), and then substituting these constants and the known general solutions of (12) into (11), we obtain the explicit solutions of nonlinear differential equation (2).

After the brief description of the methods, we now apply these for solving the Kudryashov-Sinelshchikov equation.

3. The Exact Solutions of the Kudryashov-Sinelshchikov Equation

3.1. Using 𝐺^′/𝐺-Expansion Method

Let 𝑢(𝑥,𝑡)=1−𝜑(𝜉), with 𝜉=𝑥−𝑐𝑡, 𝑐≠0, that is, 𝑢(𝑥,𝑡)=1−𝜑(𝑥−𝑐𝑡), where 𝑐 is the wave speed. Under this transformation, (1) can be reduced to the following ordinary differential equation (ODE): 𝑐𝜑^′−𝛼1−𝜑𝜑^′−𝜑^′′′+1−𝜑𝜑^′′′−𝛽𝜑^′𝜑^′′=0. (13) Integrating (13) once with respect to 𝜑 and setting the constant of integration to zero, we have 12𝛼𝜑²+()1𝑐−𝛼𝜑+2𝛼−𝜑𝜑^′′−12𝛽𝜑^′2=0. (14)

Balancing 𝜑𝜑^′′ with (𝜑^′)² in (10) we find that 𝑛+𝑛+2=2(𝑛+1), so 𝑛 is an arbitrary positive integer. For simplify, we take 𝑛=1. Suppose that (14) owns the solutions in the form 𝜑(𝜉)𝐺=𝑎0+𝑎1^′𝐺𝐺+𝑏1^′𝐺⁻¹. (15) Substituting (15) along with (5) into (14) and then setting all the coefficients of 𝐺^′/𝐺(𝑘=0,1,…) of the resulting system's numerator to zero yield a set of overdetermined nonlinear algebraic equations about 𝑎0, 𝑎1, 𝑏1, 𝑐, 𝛼, 𝛽. Solving the overdetermined algebraic equations, we can obtain the following results.

Case 1. We have 𝑎0=±−𝜆²+4𝜇⁻¹𝜆,𝑎1=0,𝑏1=±2−𝜆²+4𝜇⁻¹𝜇,𝑐=𝜆²−4𝜇,𝛼=𝜆²−4𝜇,𝛽=−4, (16)

where 𝜆, 𝜇 are arbitrary constants and 𝜆²≠4𝜇.

Case 2. We have 𝑎0=±−𝜆²+4𝜇⁻¹𝜆,𝑎1=±2−𝜆²+4𝜇⁻¹,𝑏1=0,𝑐=𝜆²−4𝜇,𝛼=𝜆²−4𝜇,𝛽=−4, (17)

where 𝜆, 𝜇 are arbitrary constants and 𝜆²≠4𝜇.

Case 3. We have 𝑊𝑎0=2𝑊,𝑎1=𝜆,𝑏1=𝜇𝑊𝜆,𝑐=6𝑊𝜇+8𝜇+𝜆²,𝛼=8𝜇+𝜆²,𝛽=−4, (18)

where 𝜆, 𝜇 are arbitrary constants, 𝜆≠0, 𝑊=RootOf(−32𝜇−4𝜆²+(16𝜇−𝜆²)𝑍²).

Using Case 3, (15) and the general solutions of (5), we can find the following travelling wave solutions of Kudryashov-Sinelshchikov equation (1).

Subcase 3.1. When Δ=𝜆²−4𝜇>0, 𝜆≠0, we obtain the hyperbolic function solutions of (1) as follows: 𝑢(𝜉)(𝜉)𝐺=1−𝜑=1−𝑎0+𝑎1^′𝐺𝐺+𝑏1^′𝐺⁻¹𝑊=1−𝜆𝜆2+𝐺^′𝐺𝐺+𝜇^′𝐺⁻¹𝑊=1−𝜆𝜆2+√Δ2𝐴₁√sinhΔ2𝜉+𝐴₂√coshΔ2𝜉×𝐴₁√coshΔ2𝜉+𝐴₂√sinhΔ2𝜉⁻¹−𝜆2√+𝜇Δ2𝐴₁√sinhΔ2𝜉+𝐴₂√coshΔ2𝜉×𝐴₁√coshΔ2𝜉+𝐴₂√sinhΔ2𝜉⁻¹−𝜆2⁻¹𝑊=1−𝜆√Δ2𝐴₁√sinhΔ2𝜉+𝐴₂√coshΔ2𝜉×𝐴₁√coshΔ2𝜉+𝐴₂√sinhΔ2𝜉⁻¹√+𝜇Δ2𝐴₁√sinhΔ2𝜉+𝐴₂√coshΔ2𝜉×𝐴₁√coshΔ2𝜉+𝐴₂√sinhΔ2𝜉⁻¹−𝜆2⁻¹, (19) where 𝜉=𝑥−(6𝑊𝜇+8𝜇+𝜆²)𝑡, Δ=𝜆²−4𝜇, 𝐴₁, 𝐴₂ are arbitrary constants.

It is easy to see that the hyperbolic function solution can be rewritten at 𝐴₁²<𝐴₂² and 𝐴₁²>𝐴₂² as follows: 𝑢()(𝜉)𝑊𝑥,𝑡=1−𝜑=1−𝜆√Δ2√tanhΔ2𝜉+𝜉₀+𝜇√Δ/2tanh√Δ/2𝜉+𝜉₀−𝜆/2,𝑢()(𝜉)𝑊𝑥,𝑡=1−𝜑=1−𝜆√Δ2√cothΔ2𝜉+𝜉₀+𝜇√Δ/2coth√Δ/2𝜉+𝜉₀−𝜆/2, (20) where, 𝜉₀=tanh⁻¹(𝐴₂/𝐴₁).

Subcase 3.2. When Δ=𝜆²−4𝜇<0, 𝜆≠0, the trigonometric function solution of (1) can be rewritten at 𝐴₁²<𝐴₂² and 𝐴₁²>𝐴₂² as follows: 𝑢()(𝜉)𝑊𝑥,𝑡=1−𝜑=1−𝜆√Δ2√tanΔ2𝜉+𝜉₀+𝜇√Δ/2tan√Δ/2𝜉+𝜉₀−𝜆/2, (21) where, 𝐴₁²<𝐴₂², 𝑢()(𝜉)𝑊𝑥,𝑡=1−𝜑=1−𝜆√Δ2cot√Δ2𝜉+𝜉₀+𝜇√Δ/2cot√Δ/2𝜉+𝜉₀−𝜆/2, (22) where, 𝐴₁²>𝐴₂².

Subcase 3.3. When Δ=𝜆²−4𝜇=0, 𝜆≠0, we obtain the rational function solutions of (1) as follows: 𝑢()(𝜉)𝑊𝑥,𝑡=1−𝜑=1−𝜆𝐴₂𝐴₁+𝐴₂𝜉𝐴+𝜇²𝐴₁+𝐴₂𝜉−𝜆2⁻¹. (23)

Using other two cases, (15), and the general solutions of (5), we could obtain other exact solutions of (1), and here we do not list all of them.

3.2. Using Generalized 𝐺^′/𝐺-Expansion Method

Suppose that (13) owns the solutions in the form 𝑢(𝜉)𝐺=𝑎0+𝑎1^′𝐺𝐺+𝑏1^′𝐺², (24) in this case, 𝐺=𝐺(𝜉) satisfies the Jacobi elliptic equation (10).

Substituting (24) along with (10) into (14) and then setting all the coefficients of 𝐺^𝑖, 𝐺^′𝐺^𝑖(𝑖=1,2,…) of the resulting system's numerator to zero yield a set of overdetermined nonlinear algebraic equations about 𝑎0, 𝑎1, 𝑏1, 𝑐, 𝛼, 𝛽. Solving the overdetermined algebraic equations, we can obtain the following results.

Case 1. We have 𝑎0=0,𝑎1=±−𝑝𝑝²,−4𝑞𝑟𝑎2=0,𝑐=4𝑝,𝛼=4𝑝,𝛽=−4. (25)

Case 2. We have 𝑎0=−𝑛−𝛼+8𝑝,𝑎1=0,𝑎2=12𝑛,𝑐=16𝑛𝑝²+192𝑛𝑟𝑞−𝛼²𝑛+𝛼,𝛼=𝛼,𝛽=−3, (26)

where, 𝑛=RootOf((48𝛼𝑝²+576𝑞𝑟𝛼−𝛼³+128𝑝³−4608𝑞𝑝𝑟)𝑍²+𝛼).

Thus using (24) and (26), the following solutions of (1) are obtained: 𝑢(𝜉)=±−𝑝𝑝²𝐺−4𝑞𝑟𝐺^′, (27) where, 𝜉=𝑥−4𝑝𝑡. Now, with the aid of Table 1, we get the following set of exact solutions of (1).

Using Case 1, (24), and the general solutions of (10), we can find the following travelling wave solutions of Kudryashov-Sinelshchikov equation (1).

Set 1.1, if 𝑟=1, 𝑝=−(𝑚²+1), 𝑞=𝑚², 𝐺(𝜉)=𝑠𝑛(𝜉), or 𝐺(𝜉)=𝑐𝑑(𝜉), then we obtain 𝑢₁(𝜉)=1−𝜑(𝜉)=1±−𝑚²𝑚²+2²𝑐𝑠(𝜉)𝑑𝑛(𝜉),𝑢₂(𝜉)=1−𝜑(𝜉)=1±−𝑚²𝑚²+2²𝑚²−1𝑠𝑑(𝜉)𝑛𝑐(𝜉), (28)(29) where, 𝜉=𝑥+4(𝑚²+1)𝑡.

When 𝑚→1, 𝑠𝑛(𝜉)→tanh(𝜉), solution (28) becomes 𝑢₁(𝜉)𝑖=1±3sech(𝜉)𝑐𝑠𝑐ℎ(𝜉), (30) where, 𝜉=𝑥+8𝑡.

Set 1.2, if 𝑟=1−𝑚², 𝑝=2𝑚²−1, 𝑞=−𝑚², 𝐺=𝑐𝑛(𝜉), then we obtain 𝑢(𝜉)=1−𝜑(𝜉)=1±𝑚²3𝑚²−2²𝑠𝑐(𝜉)𝑑𝑛(𝜉), (31) where, 𝜉=𝑥−4(2𝑚²−1)𝑡.

When 𝑚→1, 𝑐𝑛(𝜉)→sech(𝜉), solution (31) becomes 𝑢(𝜉)=1−𝜑(𝜉)=1±tanh(𝜉), (32) where, 𝜉=𝑥−4𝑡.

Set 1.3, if 𝑟=𝑚²−1, 𝑝=2−𝑚², 𝑞=−1, 𝐺(𝜉)=𝑑𝑛(𝜉), then we obtain 𝑢(𝜉)=1−𝜑(𝜉)=1±12𝑚²−3²𝑚²𝑠𝑛(𝜉)𝑐𝑑(𝜉), (33) where, 𝜉=𝑥−4(2−𝑚²)𝑡.

When 𝑚→1, 𝑑𝑛(𝜉)→sech(𝜉), solution (33) becomes 𝑢(𝜉)=1±tanh(𝜉), (34) where, 𝜉=𝑥−4𝑡. It is the same with the solution (34).

Set 1.4, if 𝑟=𝑚², 𝑝=−(𝑚²+1), 𝑞=1, 𝐺(𝜉)=𝑛𝑠(𝜉), or 𝐺(𝜉)=𝑑𝑐(𝜉), then we obtain 𝑢₁(𝜉)=1−𝜑(𝜉)=1±−12𝑚²+1²(𝜉)(𝜉),𝑢𝑑𝑠𝑐𝑛₂(𝜉)=1−𝜑(𝜉)=1±−12𝑚²+1²𝑚²−1𝑠𝑐(𝜉)𝑛𝑑(𝜉), (35) where, 𝜉=𝑥+4(𝑚²+1)𝑡.

When 𝑚→0, 𝑛𝑠(𝜉)→𝑐𝑠𝑐(𝜉), 𝑑𝑐(𝜉)→sec(𝜉), solution (35) becomes 𝑢₁(𝜉)=1±𝑖cot(𝜉),𝑢₂(𝜉)=1±𝑖tan(𝜉), (36)(37) where, 𝜉=𝑥+4𝑡.

When 𝑚→1, 𝑛𝑠(𝜉)→coth(𝜉), solution (34) becomes 𝑢(𝜉)𝑖=1±3sech(𝜉)𝑐𝑠𝑐ℎ(𝜉). (38) It is the same with solution (30). where, 𝜉=𝑥+8𝑡.

Set 1.5, if 𝑟=−𝑚², 𝑝=2𝑚²−1, 𝑞=1−𝑚², 𝐺(𝜉)=𝑛𝑐(𝜉), then we obtain 𝑢(𝜉)=1−𝜑(𝜉)=1±√−2𝑚²+1𝑑𝑛(𝜉)𝑠𝑐(𝜉), (39) where, 𝜉=𝑥+4(𝑚²+1)𝑡.

Set 1.6 if 𝑟=1/4, 𝑝=(1−2𝑚²)/2, 𝑞=1/4, 𝐺(𝜉)=𝑛𝑠(𝜉)+𝑐𝑠(𝜉), then we obtain 𝑢(𝜉)(𝜉)√=1−𝜑=1±222𝑚²−1𝑚²𝑚²(𝜉),−1𝑑𝑠 (40) where, 𝜉=𝑥−2(1−2𝑚²)𝑡.

Set 1.7 if 𝑟=(1−𝑚²)/4, 𝑝=(1+𝑚²)/2, 𝑞=(1−𝑚²)/4, 𝐺(𝜉)=𝑛𝑐(𝜉)+𝑠𝑐(𝜉), solution (39) becomes 𝑢(𝜉)=1−𝜑(𝜉)=1±−𝑚²+12𝑚²𝑑𝑐(𝜉), (41) where, 𝜉=𝑥−2(1+𝑚²)𝑡.

Similarly, we can write down the other sets of exact solutions of (1) with the help of Table 1 and the Case 2, which are omitted for convenience. Thus using the generalized form of the 𝐺^′/𝐺-expansion method, we can obtain families of the exact traveling wave solutions of (1) in terms of Jacobi elliptic functions. Under some conditions, these solutions change into hyperbolic and trigonometric functional forms.

3.3. Using Extended 𝐺^′/𝐺-Expansion Method

Suppose that (14) owns the solutions in the form 𝜑(𝜉)𝐺=𝑎0+𝑎1^′𝐺𝐺+𝑎2^′𝐺²+𝑏1𝜎11+𝜇𝐺^′𝐺²𝐺+𝑏2^′𝐺𝜎11+𝜇𝐺^′𝐺², (42) where 𝑎0, 𝑎1, 𝑎2, 𝑏1, 𝑏2 are constants to be determined later, 𝜎=±1, 𝑛 is a positive integer, and 𝐺=𝐺(𝜉) satisfies the second-order linear ODE (12).

Substituting (42) along with (12) into (14) and then setting all the coefficients of (𝐺^′/𝐺)^𝑘 and (𝐺^′/𝐺)^𝑘𝜎(1+(1/𝜇)(𝐺^′/𝐺)²)  (𝑘=0,1,…) of the resulting system to zero yield a set of overdetermined nonlinear algebraic equations about 𝑎0, 𝑎1, 𝑎2, 𝑏1, 𝑏2, 𝑐, 𝛼, 𝛽. Solving the overdetermined algebraic equations, we can obtain the following results.

Case 1. We have 𝑎0=0,𝑎1=±1𝜇,𝑎2=0,𝑏1=0,𝑏2=0,𝑐=−4𝜇,𝛼=−4𝜇,𝛽=−4. (43)

Case 2. We have 𝑎0=0,𝑎1=±1𝜇,𝑎2=0,𝑏1=±1𝛿,𝑏2=0,𝑐=−𝜇,𝛼=−𝜇,𝛽=−4. (44)

Case 3. We have 𝐾𝑎0=𝛼+8𝜇𝐾−𝛼+4𝜇,𝑎1=0,𝑎2=12,()−𝛼+4𝜇𝑏1=0,𝑏2=0,𝑐=4𝐾𝜇+𝐾+1𝛼,𝛼=𝛼,𝛽=−3, (45)

where, 𝐾=±𝛼/(𝛼+8𝜇).

Case 4. We have 𝛼𝑎0=5𝜇+𝛼rootof𝛼+2𝜇𝑍²−𝛼2𝜇²−𝛼𝜇−𝛼²,𝑎1=0,𝑎2=6RootOf𝛼+2𝜇𝑍²−𝛼,𝜇−𝛼𝑏1=0,𝑏2=6RootOf𝛼𝜎+2𝜇𝜎𝑍²−𝛼𝜇,𝛼𝜇−𝛼𝛼=𝛼,𝛽=−3,𝑐=𝜇+2𝜇RootOf𝛼+2𝜇𝑍²−𝛼+𝛼RootOf𝛼+2𝜇𝑍²×−𝛼+𝛼𝛼+2𝜇RootOf𝛼+2𝜇𝑍²−𝛼⁻¹. (46)

Using Case 1, (42), and the general solutions of (12), we can find the following travelling wave solutions of Kudryashov-Sinelshchikov equation (1).

Subcase 1.1. When 𝜇<0, we have the hyperbolic function solution as 𝑢(𝜉)(𝜉)𝐴=1−𝜑=1±𝑖¹sinh√−𝜇𝜉+𝐴₂cosh√−𝜇𝜉𝐴₁cosh√−𝜇𝜉+𝐴₂sinh√,−𝜇𝜉 (47) where, 𝜉=𝑥+4𝜇𝑡.

In particular, setting 𝐴₁=0, 𝐴₂≠0, then (47) can be written as 𝑢(𝜉)=1±𝑖coth√−𝜇𝜉. (48)

Setting 𝐴₁≠0, 𝐴₂=0, then (47) can be written as 𝑢(𝜉)=1±𝑖tanh√−𝜇𝜉. (49)

Subcase 1.2. When 𝜇>0, we have the trigonometric function solution as 𝑢(𝜉)=1−𝜑(𝜉)=1±−𝐴₁sin√𝜇𝜉+𝐴₂cos√𝜇𝜉𝐴₁cos√𝜇𝜉+𝐴₂sin√.𝜇𝜉 (50)

In particular, setting 𝐴₁=0, 𝐴₂≠0, then (50) can be written as 𝑢(𝜉)=1±cot√𝜇𝜉. (51) setting 𝐴₁≠0, 𝐴₂=0, then (50) can be written as 𝑢(𝜉)=1±tan√𝜇𝜉, (52) where, 𝜉=𝑥+4𝜇𝑡.

Using Case 2, (42), and the general solutions of (12), we can find the following travelling wave solutions of Kudryashov-Sinelshchikov equation (1).

Subcase  2.1. When 𝜇<0, we have the hyperbolic function solution as 𝑢(𝜉)(𝜉)𝐴=1−𝜑=1±𝑖¹sinh√−𝜇𝜉+𝐴₂cosh√−𝜇𝜉𝐴₁cosh√−𝜇𝜉+𝐴₂sinh√±−𝜇𝜉𝐴1−¹sinh(√−𝜇𝜉)+𝐴₂cosh(√−𝜇𝜉)𝐴₁cosh(√−𝜇𝜉)+𝐴₂sinh(√−𝜇𝜉)². (53)

In particular, setting 𝐴₁=0,𝐴₂≠0, then (53) can be written as 𝑢(𝜉)=1±𝑖coth√−𝜇𝜉±𝑐𝑠𝑐ℎ√−𝜇𝜉. (54) Setting 𝐴₁≠0, 𝐴₂=0, then (53) can be written as 𝑢(𝜉)=1±𝑖tanh√−𝜇𝜉±sech√−𝜇𝜉, (55) where, 𝜉=𝑥+𝜇𝑡.

Subcase  2.2. When 𝜇>0, we have the trigonometric function solution as 𝑢(𝜉)=1−𝜑(𝜉)=1±−𝐴₁sin√𝜇𝜉+𝐴₂cos√𝜇𝜉𝐴₁cos√𝜇𝜉+𝐴₂sin√±𝜇𝜉1+−𝐴₁sin(√𝜇𝜉)+𝐴₂cos(√𝜇𝜉)𝐴₁cos(√𝜇𝜉)+𝐴₂sin(√𝜇𝜉)². (56)

In particular, setting 𝐴₁=0, 𝐴₂≠0, then (56) can be written as 𝑢(𝜉)=1±cot√𝜇𝜉±𝑐𝑠𝑐√𝜇𝜉. (57) Setting 𝐴₁≠0, 𝐴₂=0, then (56) can be written as 𝑢(𝜉)=1±tan√𝜇𝜉±sec√𝜇𝜉, (58) where, 𝜉=𝑥+𝜇𝑡.

Similarly, we can get the other exact solutions of (1) in Cases 3 and 4, which are omitted for convenience.

Remark 1. The validity of the solutions we obtained is verified.

Remark 2. The solutions expressed by Jacobi elliptic functions are not given in the related literature. So, the solutions we obtained are new.

Remark 3. The solutions we got are general involving various arbitrary parameters. If we set the parameters to special values, some results in the literature can be obtained.

4. Conclusions

In the present work, we successfully obtained exact traveling wave solutions of the Kudryashov-Sinelshchikov equation using the 𝐺^′/𝐺-expansion method and its variants. some obtained new exact and explicit analytic solutions are in general forms involving various arbitrary parameters. These solutions are expressed by the hyperbolic functions, the trigonometric functions, the rational functions, and the Jacobi elliptic functions. The results of [1–11] have been enriched.

Acknowledgments

This research is supported by the Natural Science Foundation of of china (11161020), the National Natural Science Foundation of Yunnan Province (2011FZ193), and Research Foundation of Honghe university (10XJY120).