What makes the Peregrine soliton so special as a prototype of freak waves?
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The formation of breathers as prototypes of freak waves is studied within the framework of the classic ‘focussing’ nonlinear Schrödinger (NLS) equation. The analysis is confined to evolution of localised initial perturbations upon an otherwise uniform wave train. For a breather to emerge out of an initial hump, a certain integral over the hump, which we refer to as the “area”, should exceed a certain critical value. It is shown that the breathers produced by the critical and slightly supercritical initial perturbations are described by the Peregrine soliton which represents a spatially localised breather with only one oscillation in time and thus captures the main feature of freak waves: a propensity to appear out of nowhere and disappear without trace. The maximal amplitude of the Peregrine soliton equals exactly three times the amplitude of the unperturbed uniform wave train. It is found that, independently of the proximity to criticality, all small-amplitude supercritical humps generate the Peregrine solitons to leading order. Since the criticality condition requires the spatial scale of the initially small perturbation to be very large (inversely proportional to the square root of the smallness of the hump magnitude), this allows one to predict a priori whether a freak wave could develop judging just by the presence/absence of the corresponding scales in the initial conditions. If a freak wave does develop, it will be most likely the Peregrine soliton with the peak amplitude close to three times the background level. Hence, within the framework of the one-dimensional NLS equation the Peregrine soliton describes the most likely freak-wave patterns. The prospects of applying the findings to real-world freak waves are also discussed.
KeywordsBreathers Nonlinear Schrödinger equation Pulses in optical fibres Rogue waves Water waves
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- 7.Kharif C, Pelinovsky E, Slyunyaev A (2009) Rogue waves in the ocean. Springer, BerlinGoogle Scholar
- 15.Osborne AR, Onorato M, Serio M (2005) Nonlinear Fourier analysis of deep-water random surface waves: theoretical formulation and and experimental observations of rogue waves. 14th Aha Huliko’s Winter Workshop, vol 25. Honolulu, Hawaii, pp 16–43Google Scholar
- 20.Mei CC, Stiassnie M, Yue DK-P (2005) Theory and applications of ocean surface waves. World SciGoogle Scholar
- 22.Zakharov VE, Shabat AB (1971) Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Zh Eksp Teor Fiz 61: 118–134Google Scholar
- 24.Newell AC (1985) Solitons in physics and mathematics. SIAMGoogle Scholar
- 25.Novikov SP, Manakov SV, Pitaevskii LP, Zakharov VE (1984) Theory of solitons: the inverse scattering method (Monographs in contemporary mathematics). Springer, 1984, 292 pp. (ISBN: 0306109778)Google Scholar
- 28.Kuznetsov EA (1977) Solitons in a parametrically unstable plasma. Sov. Phys. - Dokl. (Engl. Transl.), 1977, 22, 507–508. On solitons in parametrically unstable plasma. Doklady USSR 236:575–577 (in Russian)Google Scholar
- 30.Slunyaev A (2005) Interaction of envelope soliton with a plane wave in nonlinear Schrödinger equation. Izvestia of Prochorov Academy of Engineering Sciences 14: 41–46 (in Russian)Google Scholar
- 31.Dold JW, Peregrine DH (1986) Water-wave modulation. 20th international conference on coastal engineering, Taipei, vol 1. pp 163–175Google Scholar
- 35.Landau LD, Lifshitz EM (1998) Quantum mechanics. Butterworth-Heinemannn, OxfordGoogle Scholar
- 40.Onorato M, Waseda T, Toffoli A, Cavaleri L, Gramstad O, Janssen PA, Kinoshita T, Monbaliu J, Mori N, Osborne AR, Serio M, Stansberg CT, Tamura H, Trulsen KM (2009) Statistical properties of directional ocean waves: the role of the modulational instability in the formation of extreme events. Phys Rev Lett 102(11): 114502CrossRefADSGoogle Scholar