On Exact Traveling Wave Solutions for ( 1 + 1 ) Dimensional Kaup-Kupershmidt Equation

In this present paper, the Fan sub-equation method is used to construct exact traveling wave solutions of the (1 + 1) dimensional Kaup-Kupershmidt equation. Many exact traveling wave solutions are successfully obtained, which contain solitary wave solutions, trigonometric function solutions, hyperbolic function solutions and Jacobian elliptic function periodic solutions with double periods.


Introduction
Nonlinear partial differential equations are widely used to describe complex phenomena in vary scientific fields and especially in areas of physics such as plasma, fluid mechanics, biology, solid state physics, nonlinear optics and so on.Therefore the investigation of the exact solutions to nonlinear equations plays an important role in the study of nonlinear science.Up to now, many powerful methods to seek for exact solutions to the nonlinear differential equations have been proposed.Among these are inverse scattering method [1], Lie group method [2,3], bifurcation method of dynamical systems [4][5][6], sinecosine method [7,8], tanh function method [9][10][11], homogenous balance method [12], Weierstrass elliptic function method [13].
Recently, Fan [14] presented the Fan sub-equation method which is a unified algebraic method to obtain many types of traveling wave solutions based on an auxiliary nonlinear ordinary differential equation with constant coefficients called Fan sub-equation.The important feature of Fan' method is to, without much extra effort and without considering the integrability of nonlinear equations, directly get a series of exact solutions in a uniform way, which cover all results of tanh function method, extended function method, F-expansion method, etc.This method is a powerful technique to symbolically compute traveling wave solutions of nonlinear evolution equations and is widely used by many researcher such as in [15][16][17] and by the references therein.
In this paper, we will use the Fan sub-equation method to discuss the (1+1) dimensional Kaup-Kupershmidt equation [18] which can be shown in the form 2 3 5 25 5 5 2

The Fan Sub-Equation Method
For a given nonlinear partial differential equation

 
, , , , , , 0, where u = u(x,t) is an unknown function, F usually is a polynomial in u(x,t).
To seek exact solutions of (2.1), we outline the Fan sub-equation method.The main steps are given below [14].
Step 1.By using the traveling wave transformation where c is a wave speed, we can reduce (2.1) to an ordinary differential equation in the form

 
, , , , 0, where the prime denotes the derivative with respect to  .
Step 2. Expand the solution of (2.3) in the form where a i (i = 1,2 then setting these coefficients to zero will give a set of algebraic equations with respect to a i (i = 1,2, •••, n) and c.
Step 5. Solve these algebraic equations to obtain c and i .Substituting these results into (2.4) yields to the general form of traveling wave solutions.a Step 6.For each solution to (2.5) which depends on the special conditions chosen for c j , it follows from (2.4) that the corresponding exact traveling wave solution of (2.1) can be constructed.

Exact Solutions for the (1 + 1) Dimensional Kaup-Kupershmidt Equation
The fifth order Kaup-Kupershmidt Equation (1.1) is one of the solitonic equations related to the integrable cases of the Henon-Heiles system and belongs to the completely integrable hierarchy of higher order KdV equations.Moreover the equation has infinite sets of conservation laws [19][20][21][22].Let us find the exact traveling wave solutions of the (1 + 1) dimensional Kaup-Kupershmidt equation by using the Fan sub-equation method.
The traveling wave transformation (2.2) permits us to reduce (1.1) to an ODE in the form According to Steps 1 and 2 in Section 2, by balancing and in (3.1), we obtain n + 3 = 3n − 1 and therefore give n = 2. Thus we can suppose that (3.1) has the following formal solutions where     satisfies (2.5).Substituting (3.2) and (2.5) into (3.1),collecting all terms with the same power in (0 ≤ k ≤ 5), then s simultaneous algebraic equations omitted here for the sake of brevity.Solving these algebraic equations with the help of Maple, we get the following two sets of solu-1) The first set of parameters is given by with c 2 > 0 being an arbitrary constant.
show the physical insight of these solitary wave solutions, here we take u 5 as an example.
Similar to Case 1, (1.1) has two peak-shaped solitary with c 2 > 0 being an arbitrary constant.
This in turn gives the following two wave solution of (1.1) ysical insight of the new solutions, we take u 11 as an example.Obviously the solution is a Jacobi elliptic function with two periods a the traveling o the negative x-direction nd describes f wave in with the wave velocity 2 2 2 1 1 c q p . Figure 2 shows the wave plot of the solution u 11 to (1.1) with c 2 = 0.5, k 1 = 0.9 and the initial status of u 11 .
and is an arbitrary constant.
Thus we can give two corresponding ing wave solutions of (1.1) Copyright © 2011 SciRes.AM  where and This in turn gives two corresponding periodic travelolutions ) ry coning wave s of (1.1

Conclusions and Summary
In this paper, the Fan sub-equation method has been successfully applied to obtain many traveling wave solutions of the (1 + 1) dimensional Kaup-Kuper equation.These rich results show that this method is effective and simple and a lot of solutions can be obtained ame time.It is also a promising method to solve ) and Postdoctoral Science oundation of Central South University.

Acknowledgements
coefficients to zero yields a set of tions.
c  ay obtain many are arbitrary constants.We m kinds of exact solutions deon the special va pending lues chosen for c j .triangle solutions of (1.1)

5 )
Figure the wave plot of the solution u 5 with c 2 = 1 and th ll-shaped solitary wTo1 shows e initial status of u 5 .Clearly the solution is a be ave with peak form and describes the traveling of wave in the negative x-direction.admits two following hyperbolic function solutions 0 being an arbitrary constant.

Figure 1 .
Figure 1.The plot of the peak-shaped solitary wave solution u 5 to (1.1) with c 2 = 1 and the initial status of u 5 .

Figure 2 .
Figure 2. The plot of the periodic traveling wave solution u 11 to (1.1) with c 2 = 0.5, k 1 = 0.9 and the initial status of u 11 .