Recently, S. Tang [1] studied the nonlinear dispersive variants the CH(n,n,m) of the generalized Camassa-Holm equation in (1 + 1), (2 + 1) and (3 + 1) dimensions respectively by using sine-cosine method, it is shown that this class gives compactons, conventional solitons, solitary patterns and periodic solutions.

It is the objective of this work to further complement our studies in [1] on the CH(n,n,m) equation. Our first interest in the present work being in implementing the tanh method [2,3] to stress its power in handling nonlinear equations so that one can apply it to models of various types of nonlinearity. The next interest is the determination of exact travelling wave solutions with distinct physical structures to the CH(n,2n – 1,2n,–n) given by

in (3 + 1) dimensions. Our approach depends mainly on the sine-cosine method [4], the tanh method [2,3], and the rational hyperbolic functions method [5] that have the advantage of reducing the nonlinear problem to a system of algebraic equations that can be solved by using Maple or Mathematica. As stated before, our approach depends mainly on the sine-cosine method, the extended tanh method, and the rational hyperbolic functions method. In what follows, we highlight the main steps of the proposed methods.

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.

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 [1]. The sine-cosine method, the tanh method, and the rational hyperbolic functions method were used to investigate variants of the CH(n,2n – 1,2n,–n) equations. The study revealed compactons solutions, solitary wave solutions, solitons, and periodic wave solutions for all examined variants.

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