^{1}

^{*}

^{2}

A nonlinear transformation of the Whitham-Broer-Kaup (WBK) model equations in the shallow water small-amplitude regime is derived by using a simplified homogeneous balance method. The WBK model equations are linearized under the nonlinear transformation. Various exact solutions of the WBK model equations are obtained via the nonlinear transformation with the aid of solutions for the linear equation.

The Whitham-Broer-Kaup model equations (WBK) [

where

In the present paper, we will apply a simplified homogeneous balance method to investigate the WBK model Equations (1) and (2), by this method a nonlinear transformation that from the solution for a linear equation to the solution for the WBK model equations is derived, and more type of solutions than those given in [

Considering the homogeneous balance between

where we use

Substituting (3) into the left hand sides of Equations (1) and (2), yields

In order to determine

Solving the algebraic equations we have

Substituting (6) into (3), yields

Using (5) and (6), the expressions (4) become

provided that the function

Based upon (7), (8) and (9), we come to the conclusion that inserting each solution of the linear equation (9) into (7), we can obtain the exact solution of the WBK model Equations (1) and (2), and the expressions (7) with linear Equation (9) can be looked upon as a nonlinear transformation that from the solution for linear Equation (9) to the solution for WBK model Equations (1) and (2), because every solution of linear Equation (9) under (7) is transformed into the solution of the WBK model Equations (1) and (2), therefore the WBK model Equations (1) and (2) can be linearized by the linear Equation (9), according to [

According to the superposition principle for a linear problem, the linear Equation (9) can admit many solutions, for example,

and so on., where integer

Substituting (10) into (7), we have the multiple soliton solutions of the WBK model Equations (1) and (2) as follows

If

where

Substituting (11) into (7), we have the periodic solutions in space variable

Similarly, substituting (12) into (7), we also have the periodic solutions in space variable

Substituting (13) into (7), we have rational solutions for the WBK model Equations (1) and (2)

4. Conclusion

We point out that the solutions

In this paper, the original HB is simplified by using a logarithmic function instead of the undetermined function appearing in the original HB. The nonlinear transformation that from the solution for the linear equation to the solution for the WBK model equation is derived by using the simplified HB. The WBK model equations are linearized under the nonlinear transformation. The multiple soliton solutions, periodic solutions in space variable and rational solutions of the WBK model equations are obtained in terms of solutions for the linear equation.

This work is supported in part by the Natural Science Foundation of Education Department of Henan Province of China (Grant No. 2011B110013, 12B110006) and the Doctoral Foundation of Henan University of Science and Technology (Grant No. 09001562).