APMAdvances in Pure Mathematics2160-0368Scientific Research Publishing10.4236/apm.2015.52010APM-54039ArticlesPhysics&Mathematics Rogue Wave for the Benjamin Ono Equation iliSong1*WeiChen1*ZhenhuiXu2*HanlinChen1*Applied Technology College, Southwest University of Science and Technology, Mianyang, ChinaSchool of Science, Southwest University of Science and Technology, Mianyang, China* E-mail:songlili29@163.com(IS);chenweimy@yeah.net(WC);xuzhenhui19@163.com(ZX);chenhanlin@swust.edu.cn(HC);260120150502828716 January 2015accepted 8 February 13 February 2015© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

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.

Benjamin Ono Equation Extended Homoclinic Test Method Homoclinic (Hateroclinic) Breather Limit Method Rogue Wave Solution
1. Introduction

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  . In this paper, we will consider the Benjamin Ono (BO) equation

where and are non-zero constants. The BO equation is one of the important nonlinear model in physics   . By means of traveling wave method, the exact solutions of the BO equation were obtained. Using the F-expansion method and the Jacobi elliptic function expansion method to the BO equation, a series of periodic wave solutions were got  . Based on an improved projective Riccat equation method, the traveling wave solutions of single variable were found  . Applying the bilinear method and extended homoclinic test approach  - , periodic solitary wave and doubly periodic solutions for the BO equation were obtained  .

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  , have triggered much interest in various physical branches. Rouge wave is a kind of wave that seems abnormal which is first served in the deep ocean. It always has two to three times amplitude higher than its surrounding waves and generally forms in a short time for which people think that it comes from nowhere. Rouge waves have been the subject of intensive research in oceanography   , optical fibers  - , superfluids  , Bose-Einstein condensates, financial markets and other related fields  - . In this work, we will apply the homoclinic (hateroclinic) breather limit method (HBLM)  , to seek rogue wave solution of the BO equation. We take the following four steps:

Step 1

By Painleve analysis, a transformation is made for some new and unknown function f.

Step 2

By using the transformation in step 1, original equation can be converted into Hirota’s bilinear form , where the D-operator  is defined by

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.

2. Rational Breather Wave (Rogue Wave)

The BO equation,

By Painleve analysis, let

where is unknown real function, and is the small perturbation parameter. Substituting (2) into (1) will get the following equation:

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 p1, p2, w1, w2, c1 and c2 are real constants to be determined.

Substituting Equation (8) into (7), collecting coefficients of the terms, , , and the constant, and let coefficients of these terms to zero, we get an algebraic equation

Solving Equation (9), then taking, we have

where w1, w2, c2 are some free real constants. Choosing and, we get from(10).

Substituting (10) into (8), we get

where, ,. Substituting (11) into (2) yields the solutions of (1) as follows, respectively

The solution (or) shows a new family of two-wave, breather solitary wave, which is a solitary wave and also is a periodic wave.

Substituting into the solution, it can be rewritten as follows

where (see Figure 1).

Now we consider a limit behavior of as the period of periodic wave goes to infinite, i.e.. By computing, we get the following result

where, and and as (see Figure 2).

Especially, if let, we will get, so the two breather wave solution can not be obtained, meanwhile, the rational breather wave solution (rogue wave solution) can’t also be find. The small perturbation parameter plays a huge part in finding rouge wave solution.

The figure of <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/4-5300833x56.png" xlink:type="simple"/></inline-formula> as<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/4-5300833x56.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/4-5300833x57.png" xlink:type="simple"/></inline-formula>, <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/4-5300833x56.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/4-5300833x57.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/4-5300833x58.png" xlink:type="simple"/></inline-formula>,<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/4-5300833x56.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/4-5300833x57.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/4-5300833x58.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/4-5300833x59.png" xlink:type="simple"/></inline-formula> The figure of U<sub>roguewave</sub> as<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/4-5300833x61.png" xlink:type="simple"/></inline-formula>, <inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/4-5300833x61.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/4-5300833x62.png" xlink:type="simple"/></inline-formula>,<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/4-5300833x61.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/4-5300833x62.png" xlink:type="simple"/></inline-formula><inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/4-5300833x63.png" xlink:type="simple"/></inline-formula>

Equation (15) is a rational solution of Equation (1), and it is also a breather-type solution. for fixed t as. So, the solution is a rogue wave solution which has two to three times amplitude higher than its surrounding waves and forms in a short time. One may think that whether the energy collection and superposition of breather solitary wave in many periods lead to a rogue wave or not.

3. Conclusion

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 u0 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.

Acknowledgements

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

ReferencesAblowitz, M.J. and Clarkson, P.A. (1991) Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9780511623998Korpel, A. and Banerjee, P. (1984) Heuristic Guide to Nonlinear Dispersive Wave Equation and Soliton-Type Solution. Proceedings of the IEEE, 72, 1109-1130. http://dx.doi.org/10.1109/PROC.1984.12992Fan, E. (2004) The Integrable Systems and The Computer Algebra. Science Press, Beijing.Fu. Z., Liu, S., et al. (2003) The JEFE Method and Periodic Solutions of Two Kinds of Nonlinear Wave Equations. Communications in Nonlinear Science and Numerical Simulation, 8, 67-75. http://dx.doi.org/10.1016/S1007-5704(02)00082-5Wang, Z., Li, D., et al. (2005) A Method for Constructing Exact Solutions and Application to Benjamin Ono Equation. Chinese Physics, 14, 2158-2163. http://dx.doi.org/10.1088/1009-1963/14/11/003Dai, Z., Huang, J., et al. (2005) Homoclinic Orbits and Periodic Solitons for Boussinesq Equation with Even Constraint. Chaos, Solitons & Fractals, 26, 1189-1194. http://dx.doi.org/10.1016/j.chaos.2005.02.025Dai, Z., Jiang, M., et al. (2006) Homoclinic Bifurcation for Boussinesq Equation with Even Constraint. Chinese Physics Letters, 23, 1065-1067. http://dx.doi.org/10.1088/0256-307X/23/5/001Dai, Z., Liu, J. and Li, D. (2009) Applications of HTA and EHTA to YTSF Equation. Applied Mathematics and Computation, 207, 360-364. http://dx.doi.org/10.1016/j.amc.2008.10.042Dai, Z., Li, Z., et al. (2008) Exact Homoclinic Wave and Soliton Solutions for the 2D Ginzburg-Landau Equation. Physics Letters A, 372, 3010-3014. http://dx.doi.org/10.1016/j.physleta.2008.01.015Dai, Z., Liu, Z. and Li, D. (2008) Exact Periodic Solitary-Wave Solution for KdV Equation. Chinese Physics Letters, 25, 1531-1533. http://dx.doi.org/10.1088/0256-307X/25/5/003Xu, Z.H., Xian, D.Q. and Chen, H.L. (2010) New Periodic Solitary-Wave Solutions for the Benjamin Ono Equation. Applied Mathematics and Computation, 215, 4439-4442. http://dx.doi.org/10.1016/j.amc.2009.11.009Ohta, Y. and Yang, J.K. (2012) General High-Order Rogue Waves and Their Dynamics in the Nonlinear Schr?dinger Equation. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 468, 1716-1740. Muller, P., Garrett, C. and Osborne, A. (2005) Rouge Waves. Oceanography, 18, 66-75.Kharif, C., Pelinovsky, E. and Slunyaey, A. (2009) Rogue Waves in the Ocean, Observation, Theories and Modeling. Springer, New York.Akhmediev, N., Ankiewicz, A. and Soto-Crespo, J.M. (2009) Rogue Waves and Rational Solutions of the Nonlinear Schr?dinger Equation. Physical Review E, 80, Article ID: 026601. http://dx.doi.org/10.1103/PhysRevE.80.026601? Solli, D.R., Ropers, C., Koonath, P. and Jalali, B. (2007) Optical Rogue Waves. Nature, 450, 1054-1057.http://dx.doi.org/10.1038/nature06402Bludov, V.Y., Konotop, V.V. and Akhmediev, N. (2009) Rogue Waves as Spatial Energy Concentrators in Arrays of Nonlinear Waveguides. Optics Letters, 34, 3015-3017. http://dx.doi.org/10.1364/OL.34.003015Ganshin, A.N., Efimov, V.B., Kolmakov, G.V., Mezhov-Deglin, L.P. and McClintock, P.V.E. (2008) Statistical Properties of Strongly Nonlinear Waves within a Resonator. Physical Review Letters, 101, Article ID: 065303.http://dx.doi.org/10.1103/PhysRevLett.101.065303Bludov, V.Y., Konotop, V.V. and Akhmediev, N. (2009) Matter Rogue Waves. Physical Review A, 80, Article ID: 033610. http://dx.doi.org/10.1103/PhysRevA.80.033610Montina, A., Bortolozzo, U., Residori, S. and Arecchi, F.T. (2013) Rogue Waves and Their Generating Mechanisms in Different Physical Contexts. Physics Reports, 528, 47-89. http://dx.doi.org/10.1016/j.physrep.2013.03.001Solli, D.R., Ropers, C. and Jalali, B. (2008) Active Control of Optical Rogue Waves for Stimulated Supercontinuum Generation. Physical Review Letters, 101, Article ID: 233902. http://dx.doi.org/10.1103/PhysRevLett.101.233902Yan, Z.Y. (2011) Vector Financial Rogue Waves. Physics Letters A, 375, 4274-4279.http://dx.doi.org/10.1016/j.physleta.2011.09.026Xu, Z.H., Chen, H.L. and Dai, Z.D. (2014) Rogue Wave for the (2+1)-Dimensional Kadomtsev-Petviashvili Equation. Applied Mathematics Letters, 37, 34-38. http://dx.doi.org/10.1016/j.aml.2014.05.005Hirota, R. (1985) Fundamental Properties of the Binary Operators in Soliton Theory and Their Generalizartion. In: Takeno, S., Ed., Dynamical Problem in Soliton Systems, Springer Series in Synergetiecs, Springer, Berlin, 42-49.Dai, Z.D., Liu, J. and Li, D.L. (2009) Applications of HTA and EHTA to YTSF Equation. Applied Mathematics and Computation, 207, 360-364. http://dx.doi.org/10.1016/j.amc.2008.10.042