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In the paper, the homoclinic (hateroclinic) breather limit method (HBLM) is applied to seek rogue wave solution of the Benjamin Ono equation. We find that the rational breather wave solution is just a rogue wave solution. This result shows that rogue wave can come from the extreme behavior of the breather solitary wave for (1+1)-dimensional nonlinear wave fields.

As is well known that solitary wave solutions of nonlinear evolution equations play an important role in nonlinear science fields, especially in nonlinear physical science, since they can provide much physical information and more insight into the physical aspects of the problem and thus lead to further applications [

where

In recent years, rogue waves, as a special type of nonlinear waves and also known as freak waves, monster waves, killer waves, extreme waves, abnormal waves [

Step 1

By Painleve analysis, a transformation

Step 2

By using the transformation in step 1, original equation can be converted into Hirota’s bilinear form

Step 3

Solve the above equation to get homoclinic (heteroclinic) breather wave solution by using extended homoclinic test approach (EHTA) [

Step 4

Let the period of periodic wave go to infinite in homoclinic (heteroclinic) breather wave solution, we can Obtain a rational homoclinic (heteroclinic) wave and this wave is just a rouge wave.

The BO equation,

By Painleve analysis, let

where

By means of the hirota bilinear operator, which is defined by

we will get

Putting (5) (6) into (3) implies the following bilinear equation:

In this case we choose extended homoclinic test function

where p_{1}, p_{2}, w_{1}, w_{2}, c_{1} and c_{2} are real constants to be determined.

Substituting Equation (8) into (7), collecting coefficients of the terms

Solving Equation (9), then taking

where w_{1}, w_{2}, c_{2} are some free real constants. Choosing

Substituting (10) into (8), we get

where

The solution

Substituting

where

Now we consider a limit behavior of

where

Especially, if let

Equation (15) is a rational solution of Equation (1), and it is also a breather-type solution.

In the paper, we apply the homoclinic (hateroclinic) breather limit method (HBLM) to find the BO equation’s breather solitary solution and rational breather solution. Meanwhile, rational breather solution obtained here is just a rogue wave solution of the BO equation. Furthermore, the small perturbation parameter u_{0} plays an important role in seeking rouge wave solution too. Next, we will try to use some methods to look for multi-rogue waves, such as the two-order wronskian determinant, Darboux transformation and so on.

The authors are grateful to the referee for a number of helpful suggestions to improve the paper.