^{1}

^{*}

^{1}

^{1}

In this paper, by using the sine-cosine method, the extended tanh-method, and the rational hyperbolic functions method, we study a class of nonlinear equations which derived from a fourth order analogue of generalized Camassa-Holm equation. It is shown that this class gives compactons, solitary wave solutions, solitons, and periodic wave solutions. The change of the physical structure of the solutions is caused by variation of the exponents and the coefficients of the derivatives.

Recently, S. Tang [

It is the objective of this work to further complement our studies in [

in (3 + 1) dimensions. Our approach depends mainly on the sine-cosine method [

For the three methods, we first use the wave variable to carry a PDE in two independent variables

into an ODE

Equation (2.2) is then integrated as long as all terms contain derivatives where integration constants are considered zeros.

The sine-cosine algorithm admits the use of the ansätz

or the ansätz

where are parameters that will be determined.

The standard tanh method introduced in [2,3] where the tanh is used as a new variable, since all derivatives of a tanh are represented by a tanh itself. We use a new independent variable

that leads to the change of derivatives:

We then apply the following finite expansion:

and

where M is a positive integer that will be determined to derive a closed form analytic solution.

It is appropriate to introduce rational hyperbolic functions methods where we set

where and are parameters that will be determined, and

The rational hyperbolic functions methods can be applied directly in a straightforward manner. We then collect the coefficients of the resulting hyperbolic functions and setting it equal to zero, and solving the resulting equations to determine, , and. This assumption will be used for the determination of solitons structures the CH(n, 2n – 1, 2n, –n) equations.

For the CH(n, 2n – 1, 2n, –n) equation given by (1.1), using the wave variable carries (1.1) into the ODE, respectively

where

Integrating (3.1) twice, respectively, using the constants of integration to be zero we find

Substituting (2.3) into (3.3) gives

Equation (3.4) is satisfied only if the following system of algebraic equations holds:

Solving the system (3.5) gives

The results (3.6) can be easily obtained if we also use the sine method (2.4). Combining (3.6) with (2.3) and (2.4), the following compactons solutions

and

are readily obtained, where

However, for we obtain the following solitary wave solutions

and

Using the assumptions of the tanh method (2.5)-(2.7) gives

To determine the parameter M we usually balance the linear terms of highest order in the resulting Equation (4.1) with the highest order nonlinear terms. This in turn gives

so that

To get a closed form analytic solution, the parameter should be an integer. A transformation formula

should be used to achieve our goal. This in turn transforms (3.6) to

Balancing and gives. The extended tanh method allows us to use the substitution

Substituting (4.6) into (4.5), collecting the coefficients of each power of and using Mapple to solve the resulting system of algebraic equations we obtain the following three sets:

and

Noting that

for

or

we obtain the solitary wave solutions

where

However, for

or

we obtain the periodic solutions

where

We now substitute the rational cosh

into (4.5), where

Collecting the coefficients of the like hyperbolic functions, and proceeding as before we find

The results (5.2) can be easily obtained if we also use the rational sinh method. This gives the solitons solutions

and

for, and the periodic wave solution

and the complex solution

for, where

.(5.8)

The basic goal of this work has been to extend our work on the CH(n,n,m) equation in [

This research was supported by NNSF of China (110- 61010).