Exact Solutions to the Boussinesq-Burgers Equations

A nonlinear transformation from the solution of a linear equation to the solution of the Boussinesq-Burgers equations is derived by using the simplified homogeneous balance method. Based on the nonlinear transformation and various given solutions of the linear equation, various exact solutions, including solitary wave solutions, rational solutions, the solutions containing hyperbolic functions and the solutions containing trigonometric functions, of the Boussinesq-Burgers equations are obtained.


Introduction
In the present paper, we will investigate the well-known Boussinesq-Burgers equations in the form by many authors with different methods (see [1]- [6] and references therein).
In this paper, we will propose a somewhat different method to find various exact solutions of Equations ((1) & ( 2)).First of all, a nonlinear transformation from the solution of a linear equation to the solution of Equations ((1) & ( 2)) is & ( 2)) can be obtained by the given various solutions of the linear equation.

Derivation of Nonlinear Transformation
Considering the homogeneous balance between x uu and x v in Equation (1), and between ( ) x uv and xxx u in Equation (2): according to the simplified homogeneous balance method [7] [8] [9], we can suppose that the solution of Equations (( 1) & ( 2)) is of the form where the constants A and B, as well as the function ( ) are to be determined later.Substituting (3), ( 4) into the left hand sides of Equations (( 1) & (2)), yields .
It is well known that the linear Equation ( 12) can admit an infinite many solutions, based on nonlinear transformation composed of ( 8), ( 9) and (12), we can obtain more solutions of Equations (( 1) & ( 2)), provided that more solutions of Equation ( 12) are given.

Conclusion
In this paper, the nonlinear transformation composed of ( 8), ( 9) and ( 12) for the Boussinesq-Burgers equations has been derived by using the simplified homogeneous balance method.The important role of the nonlinear transformation is that the problem of solving nonlinear Boussinesq-Burgers equations becomes that of solving a linear equation, and the latter is much easily to solve for the researchers.
α = constant.Equations ((1) & (2)) emerge in the investigation of fluid flow, and describe the proliferation of shallow water waves. of the water surface above horizontal at bottom.Equations ((1) & (2)) have been investigated