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We formulate efficient polynomial expansion methods and obtain the exact traveling wave solutions for the generalized Camassa-Holm Equation. By the methods, we obtain three types traveling wave solutions for the generalized Camassa-Holm Equation: hyperbolic function traveling wave solutions, trigonometric function traveling wave solutions, and rational function traveling wave solutions. At the same time, we have shown graphical behavior of the traveling wave solutions.

The study of dispersive waves originated from the study of water waves. To find the exact solutions of nonlinear evolution equation arising in mathematical physics plays an important role in the study of nonlinear physical phenomena. There exists an important class of solutions of nonlinear evolution equations is called traveling wave solutions which attract the interest of many mathematicians and physicists. The traveling wave solutions reduce the two variables, namely, the space variable x and the time variable t, of a partial differential equation (PDE) to an ordinary differential equation (ODE) with one independent variable

are made. In [

the four well established nonlinear evolution equation; Seadawy et al. [

In 1993, Camassa and Holm used Hamiltonian method to derive a new completely integrable shallow water wave equation

where u is the fluid velocity in the x direction (or equivalently the height of the water’s free surface above a flat bottom),

where

which is called CH-g equation. Here

In [

This paper is organized as follows. In Section 1, an introduction is presented. In Section 2, a description of the polynomial expansion method is formulated. In Section 3, the traveling wave solutions of the GCH are obtained. Finally, the paper ends with a conclusion in the Section 4.

In this section we describe the polynomial expansion methods for finding the traveling wave solutions of nonlinear evolution equation. Suppose a nonlinear equation which has independent space variable x and time variable t is given by

where

Suppose that

where “'” is the derivative with respect to

Step 1. Suppose the solution of Equation (6) can be expressed by a polynomial in

where

where

Step 2. Substituting (7) into (6). At first, balancing two highest-order, get the value of N. Then separate all

terms with same order of

Step 3. Since we can get the general solutions of Equation (8), then substituting

Step 1. Suppose the solution of Equation (6) can be expressed by a polynomial in

where

Step 2. Equating two highest-order terms in the ODE (6) and getting the value of N.

Step 3. Let the coefficients of

Step 4. By using Maple, we can solve the algebraic equations in step 2 and we obtain the traveling wave solutions of (5).

In this section, we will employ the proposed polynomial expansion methods to solve the generalized Camassa- Holm Equation (4). Substituting

where “'” is the derivative with respect to

In this section, we apply the

Balancing the terms

where

Substituting (11), (12), (13), and (14) into Equation (10), let the coefficients of

be zero, we obtain the algebraic equation system for

Solving the algebraic equation system by Maple we obtained six types of solutions:

where

where

where

where

where

where

Next, we use the solution sets from I to VI and the solutions of (8) to obtain the solutions of (10).

For I, substituting the solution set (15) and the corresponding solutions of (8) into (11), we obtain the hyperbolic function traveling wave solutions of (10) as follows:

where

For II, substituting the solution set (16) and the corresponding solutions of (8) into (11), we obtain the rational function traveling wave solutions of (10) as follows:

where

For III, substituting the solution set (17) and the corresponding solutions of (8) into (11), we obtain the traveling wave solutions of (10) as follows:

When

where

When

where

For IV, when

When

For V and VI, we have the rational function traveling wave solutions of (10) like (22).

In addition, the figures of IV are similar to the figures of III, and the figures of V and VI are similar to the figure of II.

In this section, we apply the sinh-tanh polynomial expansion method to solve the Equation (10).

Balancing the terms

where

Substituting (25), (26), (27), and (28) into Equation (10), let the coefficients of

where

where c and

where

where

Therefore, we obtain the solutions of (10) by the solution sets from case 1 to case 4.

For i, substituting the solution set (29) into (11), we obtain the hyperbolic function traveling wave solutions of (10) as follows:

where

For ii, substituting the solution set (30) into (11), we obtain the hyperbolic function traveling wave solutions of (10) as follows:

where

For iii, substituting the solution set (31) into (11), we obtain the hyperbolic function traveling wave solutions of (10) as follows:

where

For iv, substituting the solution set (32) into (11), we obtain the hyperbolic function traveling wave solutions

of (10) as follows:

where

We proposed efficient polynomial expansion methods and obtained the exact traveling wave solutions of generalized Camassa-Holm equation. By polynomial expansion method we obtain hyperbolic function traveling wave solutions, trigonometric function traveling wave solutions, and rational function traveling wave solutions. On comparing with the polynomial expansion methods and other methods to find out the traveling wave for PDEs, the polynomial expansion methods are more effective, powerful and convenient. Moreover, the polynomial expansion methods can be used to solve any high-order degree PDEs.

The research is supported in part by the Science and Research Foundation of Yunnan Province Department of Education under grant No. 2015Y277, in part by the Natural Science Foundation of China under grant No. 11161038 and in part by Yunnan Province and Shanghai University of Finance and Economics Education Cooperation consulting Project under grant No. 42111217003.

Junliang Lu,Xiaochun Hong, (2016) Exact Traveling Wave Solutions for Generalized Camassa-Holm Equation by Polynomial Expansion Methods. Applied Mathematics,07,1599-1611. doi: 10.4236/am.2016.714138