Abstract

Exact traveling wave solutions of the Kudryashov-Sinelshchikov equation are studied by the -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, where , 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, 35]. 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 and 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 [810]. In [11], the equation is studied by the Lie symmetry method.

The -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 -expansion method, the modified -expansion method, the extended -expansion method, and the improved -expansion method have been proposed in [1315], respectively. The aim of this paper is to derive more and new traveling wave solutions of the Kudryashov-Sinelshchikov equation by the -expansion method and its variants.

The organization of the paper is as follows: in Section 2, a brief account of the -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 -Expansion Method

Step 1. Consider a general nonlinear PDE in the form Using , , we can rewrite (2) as the following nonlinear ODE: where the prime denotes differentiation with respect to .

Step 2. Suppose that the solution of ODE (3) can be written as follows: where , are constants to be determined later, is a positive integer, and satisfies the following second-order linear ordinary differential equation: where , are real constants. The general solutions of (5) can be listed as follows.

When , we obtain the hyperbolic function solution of (5)

When , we obtain the trigonometric function solution of (5)

When , we obtain the solution of (5) where and are arbitrary constants.

Step 3. Determine the positive integer 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 of the resulting system's numerator to zero yields a set of over-determined nonlinear algebraic equations for and .

Step 5. Assuming that the constants and 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 -Expansion Method

In generalized version [13], one makes an ansatz for the solution as where satisfies the following Jacobi elliptic equation: where , , and are the arbitrary constants to be determined later and . Substituting (9) into (3) and using (10), we obtain a polynomial in , . Equating each coefficient of the resulted polynomials to zero yields a set of algebraic equations for , , , and . Now, substituting 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).

2.3. The Extended -Expansion Method

In the extended form of this method [15], the solution of (3) can be expressed as where , , are constants to be determined later, , is a positive integer, and satisfies the following second order linear ODE: where is a constant. Substituting (11) into (3), using (12), collecting all terms with the same order of and together, and then equating each coefficient of the resulting polynomial to zero yield a set of algebraic equations for , , , . On solving these algebraic equations, we obtain the values of the constants , , , , 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 -Expansion Method

Let , with , , that is, , where is the wave speed. Under this transformation, (1) can be reduced to the following ordinary differential equation (ODE): Integrating (13) once with respect to and setting the constant of integration to zero, we have

Balancing with in (10) we find that , so is an arbitrary positive integer. For simplify, we take . Suppose that (14) owns the solutions in the form Substituting (15) along with (5) into (14) and then setting all the coefficients of of the resulting system's numerator to zero yield a set of overdetermined nonlinear algebraic equations about , , , , , . Solving the overdetermined algebraic equations, we can obtain the following results.

Case 1. We have

where , are arbitrary constants and .

Case 2. We have

where , are arbitrary constants and .

Case 3. We have

where , are arbitrary constants, , .

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 , , we obtain the hyperbolic function solutions of (1) as follows: where , , , are arbitrary constants.

It is easy to see that the hyperbolic function solution can be rewritten at and as follows: where, .

Subcase 3.2. When , , the trigonometric function solution of (1) can be rewritten at and as follows: where, , where, .

Subcase 3.3. When , , we obtain the rational function solutions of (1) as follows:

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 -Expansion Method

Suppose that (13) owns the solutions in the form in this case, satisfies the Jacobi elliptic equation (10).

Substituting (24) along with (10) into (14) and then setting all the coefficients of , of the resulting system's numerator to zero yield a set of overdetermined nonlinear algebraic equations about , , , , , . Solving the overdetermined algebraic equations, we can obtain the following results.

Case 1. We have

Case 2. We have

where, .

Thus using (24) and (26), the following solutions of (1) are obtained: where, . 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 , , , , or , then we obtain where, .

When , , solution (28) becomes where, .

Set 1.2, if , , , , then we obtain where, .

When , , solution (31) becomes where, .

Set 1.3, if , , , , then we obtain where, .

When , , solution (33) becomes where, . It is the same with the solution (34).

Set 1.4, if , , , , or , then we obtain where, .

When , , , solution (35) becomes where, .

When , , solution (34) becomes It is the same with solution (30). where, .

Set 1.5, if , , , , then we obtain where, .

Set 1.6 if , , , , then we obtain where, .

Set 1.7 if , , , , solution (39) becomes where, .

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 -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 -Expansion Method

Suppose that (14) owns the solutions in the form where , , , , are constants to be determined later, , is a positive integer, and satisfies the second-order linear ODE (12).

Substituting (42) along with (12) into (14) and then setting all the coefficients of and    of the resulting system to zero yield a set of overdetermined nonlinear algebraic equations about , , , , , , , . Solving the overdetermined algebraic equations, we can obtain the following results.

Case 1. We have

Case 2. We have

Case 3. We have

where, .

Case 4. We have

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 , we have the hyperbolic function solution as where, .

In particular, setting , , then (47) can be written as

Setting , , then (47) can be written as

Subcase 1.2. When , we have the trigonometric function solution as

In particular, setting , , then (50) can be written as setting , , then (50) can be written as where, .

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 , we have the hyperbolic function solution as

In particular, setting , then (53) can be written as Setting , , then (53) can be written as where, .

Subcase  2.2. When , we have the trigonometric function solution as

In particular, setting , , then (56) can be written as Setting , , then (56) can be written as where, .

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 -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 [111] 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).