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Under the travelling wave transformation, some nonlinear partial differential equations such as the Getmanou equation are transformed to ordinary differential equation. Then using trial equation method and combing complete discrimination system for polynomial, the classifications of all single traveling wave solution to this equation are obtained.

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 [

Recently, Professor Liu proposed a powerful method named trial equation method [

The Getmanou equation [

Or equivalently

Taking the traveling wave transformation

we take the trial equation as follows

According to the trial equation method of rank homogeneous equation, balancing

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

When

Then Equation (6) becomes

where

When

Then Equation (6) becomes

where

We write the complete discrimination system for polynomial

Then we consider the following ODE

where

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

Case 1.

where

When

The corresponding solution is

Case 2.

When

The corresponding solution is

Case 3.

where

When

(i) If

The corresponding solution is

(ii) If

The corresponding solution is

Case 4.

where

When

(i) If

The corresponding solution is

(ii) If

The corresponding solution is

(iii) If

The corresponding solution is

When

(i) If

The corresponding solution is

(ii) If

The corresponding solution is

(iii) If

The corresponding solution is

Case 5.

where

When

The corresponding solution is

When

The corresponding solution is

Case 6.

where

When

The corresponding solution is

where

Case 7.

where

When

(i) If

The corresponding solution is

(ii) If

The corresponding solution is

When

(i)

The corresponding solution is

(ii)

The corresponding solution is

where

Case 8.

where

When

The corresponding solution is

When

The corresponding solution is

where

We choose

Case 9.

where

When

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

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.