Application of Trial Equation Method for Solving the Getmanou Equation

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.


Introduction
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.

Application of Trial Equation Method
The Getmanou equation [12] reads as ( ) Or equivalently ( ) ( ) Taking the traveling wave transformation ( ) u u ξ = and 1 kx t ξ ω = + , we can obtain the corresponding ODE ( ) ( ) we take the trial equation as follows According to the trial equation method of rank homogeneous equation, balancing 2 u u′′ with 5 u gets 3 m = .
Let the coefficient ( ) 0,1, 2,3, 4,5 i r i = be zero, we will yield nonlinear algebraic equations.Solving the equations, we will determine the values of 0 1 2 3 , , , , a a a a d .We get When 3 > 0 a , we take transformations as follows Then Equation ( 6) becomes . w w pw qw r where w is a function of ξ and When 3 < 0 a , we take transformations as follows Then Equation ( 6) becomes . w w pw qw r where w is a function of ξ and We write the complete discrimination system for polynomial ( )

Classification of the Traveling Wave Solutions
According to the complete discrimination system for the fouth order polynomial, we give the corresponding single traveling wave solutions to Equation (1).Case 1. 4 0 where 1 l and 1 s are real numbers, 1 > 0 s .When 1 ε = , we have ( ) The corresponding solution is ( ) Then we have ( ) 4 .
The corresponding solution is ( ) where α and β are real numbers, > The corresponding solution is ( ) The corresponding solution is ( ) where α , β , γ are real numbers, and The corresponding solution is ( ) The corresponding solution is The corresponding solution is The corresponding solution is ( ) The corresponding solution is The corresponding solution is where α and β are real numbers.
When 1 The corresponding solution is ( ) The corresponding solution is ( ) where α , 1 l and 1 s are real numbers.When 1 ε = , we have The corresponding solution is where ( ) where 1 α , 2 α , 3 α and 4 α are real numbers, and 1 The corresponding solution is ( The corresponding solution is The corresponding solution is The corresponding solution is where )( ) where α , β , 1 l , 1 s are real numbers, and > α β , 1 > 0 The corresponding solution is When 1 ε = − , we have ( ) ( ) ( ) ( ) The corresponding solution is where . 1 We  The corresponding solution is where ( )  64) and (68), the integration constant 0 ξ has been rewritten, but we still use it.The classifications of all single traveling wave solution to this equation are obtained.

Conclusion
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.