Many problems in natural and engineering sciences are modeled by partial differential equations (PDE). Looking for the solutions of the equation, especially the exact solutions, is very important. These exact solutions can describe many important phenomena in physics and other fields and also help physicists to understand the mechanisms of the complicated physical phenomena. Many mathematicians and physicists work in the field, and a variety of powerful methods have been employed to study nonlinear phenomena, such as the inverse scattering transform [1] , the Bäcklund transformation method [2] , the Darboux transformation [3] , the homogeneous balance method [4] , the tanh function method [5] , the exp-function method [6] , the G'/G-expansion method [7] , and so on.

Recently, Professor Liu proposed a powerful method named trial equation method [8] [9] for finding exact solutions to nonlinear differential equations. In this paper, I mainly use Liu’s trial equation method and the theory of complete discrimination system for the fouth-order polynomial [10] -[12] to solve exact solutions of the Getmanou equation which has already been solved based on discrimination system of the fifth-order polynomial by Fan [13] . But as we can see, the solving process is very simple and clear with the trial equation method combined with complete discrimination system for polynomial.

The Getmanou equation [12] reads as

Or equivalently

Taking the traveling wave transformation and, we can obtain the corresponding ODE

we take the trial equation as follows

According to the trial equation method of rank homogeneous equation, balancing with gets.

Equation (4) has the following specific form

From Equation (5), we get

Substituting Equations (5) and (6) into Equation (3), we have

where

Let the coefficient be zero, we will yield nonlinear algebraic equations. Solving the equations, we will determine the values of. We get

When, we take transformations as follows

Then Equation (6) becomes

where is a function of and

When, we take transformations as follows

Then Equation (6) becomes

where is a function of and

We write the complete discrimination system for polynomial as follows

Then we consider the following ODE

where. Rewrite Equation (17) by integral form as follows

According to the complete discrimination system for the fouth order polynomial, we give the corresponding single traveling wave solutions to Equation (1).

Case 1., ,. Then we have

where and are real numbers,.

When, we have

The corresponding solution is

Case 2., ,. Then we have

When, we have

The corresponding solution is

Case 3., , ,. Then we have

where and are real numbers,.

When

(i) If or, we have

The corresponding solution is

(ii) If, we have

The corresponding solution is

Case 4., ,. Then we have

where, , are real numbers, and.

When

(i) If, or if, , we have

The corresponding solution is

(ii) If, or if, , we have

The corresponding solution is

(iii) If, we have

The corresponding solution is

When

(i) If, or if, , we have the corresponding solution is

The corresponding solution is

(ii) If, or if, , we have

The corresponding solution is

(iii) If, we have

The corresponding solution is

Case 5., , ,. Then we have

where and are real numbers.

When, if, or if, , we have

The corresponding solution is

When, if, or if, , we have

The corresponding solution is

Case 6.,. Then we have

where, and are real numbers.

When, we have

The corresponding solution is

where.

Case 7., ,. Then we have

where, , and are real numbers, and.

When

(i) If or, we have

The corresponding solution is

(ii) If, we have

The corresponding solution is

When

(i), we have

The corresponding solution is

(ii), we have

The corresponding solution is

where.

Case 8.,. Then we have

where, , , are real numbers, and,.

When, we have

The corresponding solution is

When, we have

The corresponding solution is

where

We choose such that.

Case 9.,. Then we have

where, , , are real numbers, and.

When, we have

The corresponding solution is

where

In Equations (21) (24) (27) (29) (32) (34) (36) (38) (40) (42) (45) (47) (50) (53) (55) (57) (59) (62) (64) and (68), the integration constant has been rewritten, but we still use it. The classifications of all single traveling wave solution to this equation are obtained.

In this paper, the trial equation method combined with complete discrimination system for polynomial has been effectively used to solve the Getmanou equation. The obtained results emphasize that the method is completely useful. With the same method, some of other equations can be dealt with.

I would like to thank the referees for their valuable suggestions.