_{1}

^{*}

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.

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 [

The Benjamin Ono equation reads as

where

we take the trial equation as follows:

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

Integrating the Equation (3)once with respect to

By Equation (3) and Equation(4), we derive the following equation

Substituting Equations (3)-(5) into Equation (2), we have

where

Let the coefficient

We get

Let

Then Equation(4) becomes

where

Denote

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

Case 1.

when

Case 2.

The corresponding solution is

Case 3.

when

According to the Equation(12), we have

where

The corresponding solutions is

when

The corresponding solutions is

where

Case 4.

when

According to the Equation(13), we have

where

The corresponding solutions is

In Equations (17), (18), (19), (21), (26), (28) and (33), the integration constant

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.

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

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[^{2}^{ }+ q(u) = 0 and Applications to Classifications of All Single Travelling Wave Solutions to Some Nonlinear Mathematical Physics Equations,” Communications in Theoretical Physics, Vol. 49, No. 2, 2008, pp. 291-296. http://dx.doi.org/10.1088/0253-6102/49/2/07

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