Application of Trial Equation Method for Solving the Benjamin Ono Equation

In the article, the nonlinear equation is reduced to an ordinary differential equation under the travelling wave transformation. Using trial equation method, the ODE is reduced to the elementary integral form. In the end, complete discrimination system for polynomial is used to solve the corresponding integrals and obtain the classification of all single travelling wave solutions to the equation.


Introduction
Nonlinear phenomena are general problems in every field of engineering technology, science research, natural world and human society activities.So the investigation of exact solutions of nonlinear equations plays a important role not only in theoretic research but in application.To obtain the travelling wave solutions, many methods were attempted, such as the inverse scattering method [1], Hirotas bilinear transformation [2,3], the tanh method [4], sine-cosine method [5], homogeneous balance method [6,7], exp-function method [8], and so on.These methods derived many solutions to most nonlinear evolution equations.Recently, Professor Liu proposed a powerful method named trial equation method for finding exact solutions to nonlinear differential equations [9][10][11].By using his method, the nonlinear differential equation is reduced to an ordinary differential equation under the travelling wave transformation.Using the trial equation method, the ODE is reduced to the elementary integral form.In the end, the complete discrimination system for polynomial is used to solve the corresponding integrals.We can obtain the classification of all single travelling wave solutions [12][13][14][15][16] to the equation.This idea is so good that many types of nonlinear differential equations can be solved by it.Using the trial equation method and complete discrimination system for polynomial, we have obtained a lot of new solutions to many nonlinear differential equations.As an application, some new solutions to the Benjamin Ono equation are given.

Application of the Trial Equation Method
The Benjamin Ono equation reads as where , β γ are parameters.Taking the traveling wave transformation ( ) Integrating the Equation (3)once with respect to 1 ξ , we get ( ) By Equation ( 3) and Equation( 4), we derive the following equation ( ) Substituting Equations ( 3)-( 5) into Equation ( 2), we have where ( ) ( ) Let the coefficient be zero,we will yield nonlinear algebraic equations.Solving the equations, we will determine the values of 0 1 2 , , , a a a d .We get satisfied, we use the complete discrimination system for the third order polynomial and have the following solving process.Let Then Equation( 4) becomes ( ) where v is a function of ξ .The integral form of Equation( 12) is ( ) ( ) to Equation(1).Case 1.
( ) < ⋅ = has a double real root and a simple real root.Then we have ( ) ( ) ( ) when 2 v λ > , the corresponding solutions are ( ) ( ) ( ) ( ) = ⋅ = has a triple root.Then we have ( ) ( ) The corresponding solution is ( ) ( ) < ⋅ = has three different real roots.Then we have ( ) ( )( )( ) < < , we take the transformation as follows According to the Equation( 12), we have ( ) ( ) .On the basis of the Equation(24) and the definition of the Jacobi elliptic sine function, we have ( ) ( ) The corresponding solutions is ( ) ( ) The corresponding solutions is ( ) = has only a real root.Then we have tan .
According to the Equation( 13), we have ) ( ) . On the basis of the Equation(31) and the definition of the Jacobi elliptic cosine function, we have The corresponding solutions is In Equations ( 17), ( 18), ( 19), ( 21), ( 26), ( 28) and (33), the integration constant 0 ξ has been rewritten,but we still use it.The solutions ( ) 1, , 7 i u i =  are all possible exact traveling wave solutions to Equation (1).It is easy to write the corresponding solutions to the Benjamin Ono equation.For brevity, we omitted.

Conclusion
Trial equation method is a systematic method to solve nonlinear differential equations.The advantage of this method is that we can deal with nonlinear equations with linear methods.This method has the characteristics of simple steps and clear effectivity.Based on the idea of the trial equation method and the aid of the computerized symbolic computation, some exact traveling wave solutions to the Benjamin Ono equation have been obtained.With the same method, some of other equations can be dealt with.
d are two arbitrary constants.When the above conditions are