Exact Solutions of the Kudryashov-sinelshchikov Equation by Modified Exp-function Method

Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, the modified exp-function method is used to seek generalized wave solutions of Kudryashov-Sinelshchikov equation. As a result , some new types of exact traveling wave solutions for arbitrary α, β are obtained which include exponential function, hyperbolic function and trigonometric function. The related results are extend. Obtained results clearly indicate the reliability and efficiency of the modified exp-function method.


Introduction
The investigation of exact solutions of nonlinear wave equations plays an important role in the study of nonlinear physical phenomena.Recently, many effective methods for obtaining exact solutions of nonlinear wave equations have been proposed, such as bäcklund transformation method, homogeneous balance method, bifurcation method, Hirotas bilinear method, the hyperbolic tangent function expansion method, the Jacobi elliptic function expansion method , F-expansion method and so on.He and Wu [1] developed the exp-function method to seek the solitary, periodic and compaction like solutions of nonlinear differential equations.It is an effective and simple method and is widely used.Based on this method, modified exp-function expansion method is proposed.
In 2010, Kudryashov and Sinelshchikov [2] 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, α, β are real parameters.In [3], 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.Eq.( 1) is called Kudryashov-Sinelshchikov equation, it is generalization of the KdV and the BKdV equation and similar but not identical to the Camassa-Holm (CH) equation, it has been studied by some authors [2,[4][5][6].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 [5].Ryabov [6] obtained some exact solutions for β = −3 and β = −4 using a modification of the truncated expansion method [8].Li and He discussed the equation by the bifurcation method of dynamical systems and the method of phase portraits analysis [9][10][11].In [12], the equation is studied by the Lie symmetry method.In [13], the equation is studied by G G -expansion method and its variants.The aim of this work is to further complement the studies on the Kudryashov-Sinelshchikov equations by the new method.
The organization of the paper is as follows: in Section 2, a brief description of the modified Exp-function method is given.In section 3, we will study the Kudryashov-Sinelshchikov equation by the new methods.Finally conclusions are given in Section 4.

Modified Exp-function method
The exp-function method was first proposed by He and Wu to solve differential equations [1].In this paper, we will introduce a modified Exp-function method.The main procedures of this method are as follows.
We consider a general nonlinear PDE in the form Using a transformation ξ = x − ct, where c are constants, we can rewrite Eq.( 2) into the following nonlinear ODE: where the prime denotes the derivation with respect to ξ.Let u = v + s, where s are constants.Then Eq.( 3) becomes Assume that the solution of Eq.( 2) can be expressed in the following form where g = e kξ which is the solution of the homogeneous linear equation corresponding to equation ( 4).In order to determine the constants c, p and q, d, we can balance the linear term of the highest order in Eq.( 3) with the highest order nonlinear term.Substituting Eq.( 5) into Eq.(3), we can get polynomial equation on g.Let the coefficient of g i be zero, and solve the equation set, the a i , b i can be determined.

Solutions of Kudryashov-Sinelshchikov equation
Using the transformation ξ = x − ct, equation (1) can be rewrite as Integrating (6) once and making the integral constant be zero, we have Let u = v + s, equation ( 9) can be rewrite as Let 1 2 α s 2 − cs = 0, then s = 0 or s = 2 c α .According to homogeneous balance principle, we get p = c and q = d.Here, we take p = c = 2, q = d = 2. Case 1 s = 0.
The solution of the linear equation corresponding to equation ( 8) is So,we can assume Substituting Eq.( 9) and Eq.( 10) into Eq.(7) yields a set of algebraic equations for g i (i = 0, 1, . . ., 16).Letting the coefficients of these terms g i to be zero yields a set of over-determined algebraic equations.Solving the system of algebraic equations, we obtain Set 1.1 Substituting ( 11) into (10), when α = c (β + 2), we can obtain following solutions of the Kudryashov-Sinelshchikov equation as follow.
If c < 0, (14) can be reduced as If c < 0, ( 14) can be reduced as . The solution of the linear equation corresponding to equation ( 8) is So,we can assume Substituting Eq.( 20) and Eq.( 19) into Eq.(7) yields a set of algebraic equations for g i (i = 0, 1, . . ., 16).Letting the coefficients of these terms g i to be zero yields a set of over-determined algebraic equations.Solving the system of algebraic equations, we obtain Set 2.1 Substituting ( 21) into (20), when α = −c β, we can obtain following solutions of the Kudryashov-Sinelshchikov equation as follow.
where g = e If c < 0, (26) can be reduced as If c < 0, (26) can be reduced as Remark 1.We can also get other similarly solutions of Equation ( 1) which are omitted for convenience.
Remark 2. The validity of the solutions we obtained are verified.Remark 3. We discussed the situation for arbitrary α and β, so the results of the related literature are extended.