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

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.


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,[3][4][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  = −3 and  = −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][9][10].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 [13][14][15], 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.
Step 2. Suppose that the solution of ODE (3) can be written as follows: where ,   ( = −, −+1, . ..) 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.
Step 3. Determine the positive integer  by balancing the highest order derivatives and nonlinear terms in (3).
Step 5. Assuming that the constants  and   ( = −, − + 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   /-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   ̸ = 0. Substituting (9) into (3) and using (10), we obtain a polynomial in   ,     ( = 1, 2, . ..).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  0 ,   ,   ( = 1, 2, . . ., ) are constants to be determined later,  = ±1,  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 2 ) together, and then equating each coefficient of the resulting polynomial to zero yield a set of algebraic equations for ,  0 ,   ,   ( = 1, . . ., ).On solving these algebraic equations, we obtain the values of the constants ,  0 ,   ,   ( = 1, . . ., ), 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.

The Exact Solutions of the Kudryashov-Sinelshchikov Equation
Integrating ( 13) once with respect to  and setting the constant of integration to zero, we have Balancing   with (  ) 2 in (10) we find that  +  + 2 = 2( + 1), so  is an arbitrary positive integer.For simplify, we take  = 1.Suppose that (14) owns the solutions in the form Substituting (15) along with (5) into (14) and then setting all the coefficients of   / ( = 0, 1, . ..) of the resulting system's numerator to zero yield a set of overdetermined nonlinear algebraic equations about 0, 1, 1, , , .Solving the overdetermined algebraic equations, we can obtain the following results.
Using Case 1, (24), and the general solutions of (10), we can find the following travelling wave solutions of Kudryashov-Sinelshchikov equation (1).
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.