Freak (or rogue) waves are the nonlinear waves of extreme amplitude, which appear from relatively calm wave background and fleetingly disappear. In the ocean, they can destroy ships, oil platforms and, most importantly, be a danger for people's life. For a long time freak waves were only the subject of sailor's mysterious stories. Active scientific study of this phenomenon started about 20-30 years ago. Nowadays there are no doubts in their existence: the ocean freak waves are well documented. Moreover, freak waves were found in optics, plasma and even in Bose-Einstein condensate. Thereby, freak waves are the universal phenomena for nonlinear wave physics.
The theory of freak waves is not complete, however there is a point of common agreement that freak waves appear because of modulation instability (MI) of a background wave. The purpose of the present talk is to provide a short review of the problem and to tell about the recent results on this subject of the Laboratory of nonlinear wave processes at Novosibirsk State University.
The most exact approach to study the aftereffects of MI development is based on the use of the exact fully nonlinear Euler equations. V.E. Zakharov and A.I. Dyachenko [1] proved the existence of the "giant breather" in the frame of the Euler equations and offered to identify freak waves with it. Another way is based on the use of envelope equations such as nonlinear Schrodinger equation (NLSE), which are expansions in powers of small parameter of the Euler equation. Certainly, this approach is restricted by choosing the value of small parameter, however the envelope equations are significantly easier for analyzing, especially NLSE which can by fully integrable by the inverse scattering transform. The alternative point of view on the formation of freak waves is associated with the NLSE. D. H. Peregrine found an intriguing pure homoclinic solution, which now called the Peregrine breather in 1983. It is a solution, which appears once in space and time from continuous wave background, and reaches three amplitudes of the initial wave. Then multi-Peregrine breathers solutions were found and nowadays it is the extensively studied field of science.
The main part of the talk will focus on the new solitonic scenario of modulation instability in the frame of the NLSE which was found by A.A. Gelash and V.E. Zakharov [2,3], studies of breathers collisions in the frame of Zakharov's equation done by A.I. Dyachenko, D.I. Kachulin and V.E. Zakharov [4], and the freak wave statistics results of D. S. Agafontsev and V.E. Zakharov [5,6].
1. A.I. Dyachenko, V.E. Zakharov. On the formation of freak waves on the surface of deep water, JETP Lett., 88(5), 307-311 (2008).
2. V.E. Zakharov and A.A. Gelash.Nonlinear Stage of Modulation Instability, Phys. Rev. Lett. 111, 054101 (2013)
3. A.A. Gelash and V.E. Zakharov.Superregularsolitonic solutions: a novel scenario of the nonlinear stage of Modulation Instability, Nonlinearity 27 (2014) R1–R39.
4. Dyachenko, A. I., Kachulin, D. I., and Zakharov, V. E. Collisions of two breathers at the surface of deep water, Nat. Hazards Earth Syst. Sci., 13, 3205-3210.
5. D.S. Agafontsev, V.E. Zakharov. Oscillatory dynamics of the classical Nonlinear Schrodinger equation, to be submitted.
6. Dmitry Agafontsev, Vladimir Zakharov. Rogue waves statistics in the framework of one-dimensional Generalized Nonlinear Schrodinger Equation, arXiv: 1202.5763.