Advertisement

Journal of Engineering Mathematics

, Volume 67, Issue 1–2, pp 11–22 | Cite as

What makes the Peregrine soliton so special as a prototype of freak waves?

  • Victor I. Shrira
  • Vladimir V. Geogjaev
Article

Abstract

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.

Keywords

Breathers Nonlinear Schrödinger equation Pulses in optical fibres Rogue waves Water waves 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Benjamin TB, Feir JE (1967) The disintegration of wavetrains in deep water. Part 1. J Fluid Mech 27: 417–430zbMATHCrossRefADSGoogle Scholar
  2. 2.
    Benney DJ, Roskes GJ (1969) Wave instabilities. Stud Appl Math 48: 377–385zbMATHGoogle Scholar
  3. 3.
    Peregrine DH (1983) Water waves, nonlinear Schrödinger equations and their solutions. J Aust Math Soc Ser B 25: 16–43zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Osborne AR, Onorato M, Serio M (2000) The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains. Phys Lett A 275: 386–393zbMATHCrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Dyachenko AI, Zakharov VE (2005) Modulational instability of Stokes waves → freak wave. JETP Lett 81: 255–259CrossRefADSGoogle Scholar
  6. 6.
    Kharif C, Pelinovsky E (2003) Physical mechanisms of the rogue wave phenomenon. Eur J Fluid Mech B/Fluids 22: 603–634zbMATHCrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Kharif C, Pelinovsky E, Slyunyaev A (2009) Rogue waves in the ocean. Springer, BerlinGoogle Scholar
  8. 8.
    Song J-B, Banner ML (2002) On determining the onset and strength of breaking for deep water waves. Part I: unforced irrotational wave groups. J Phys Oceanogr 32: 2541–2558MathSciNetGoogle Scholar
  9. 9.
    Tanaka M (1990) Maximum amplitude of modulated wave train. Wave Motion 12: 559–568CrossRefGoogle Scholar
  10. 10.
    Alber IE (1978) The Effects of Randomness on the Stability of Two-Dimensional Surface Wave trains. Proc R Soc Lond A 363(1715): 525–546zbMATHCrossRefMathSciNetADSGoogle Scholar
  11. 11.
    Janssen PAEM (1983) Long-time behaviour of a random inhomogeneous field of weakly nonlinear surface gravity waves. J Fluid Mech 133: 113–132. doi: 10.1017/S0022112083001810 zbMATHCrossRefMathSciNetADSGoogle Scholar
  12. 12.
    Stiassnie M, Regev A, Agnon Y (2008) Recurrent solutions of Alber’s equation for random water-wave fields. J Fluid Mech 598: 245–266zbMATHCrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Janssen PAEM (2003) Nonlinear four-wave interactions and freak waves. J Phys Oceanogr 33: 863–884CrossRefMathSciNetADSGoogle Scholar
  14. 14.
    Annenkov S, Shrira V (2009) Evolution of kurtosis for wind waves. Geophys Res Lett 36: L13603. doi: 10.1029/2009GL038613 CrossRefADSGoogle Scholar
  15. 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
  16. 16.
    Islas LA, Schober CM (2005) Predicting rogue waves in random oceanic sea states. Phys Fluids 17: 0317011-4CrossRefMathSciNetGoogle Scholar
  17. 17.
    Slunyaev A, Pelinovsky E, Guedes Soares C (2005) Modelling freak waves from the North Sea. Appl Ocean Res 27: 12–22CrossRefGoogle Scholar
  18. 18.
    Slunyaev A (2006) Nonlinear analysis and simulations of measured freak wave time series. Eur J Mech -B Fluids 25: 621–635zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Akhmediev N, Ankiewicz A, Taki M (2008) Waves that appear from nowhere and disappear without a trace. Phys Lett A 373: 675–678CrossRefADSGoogle Scholar
  20. 20.
    Mei CC, Stiassnie M, Yue DK-P (2005) Theory and applications of ocean surface waves. World SciGoogle Scholar
  21. 21.
    Janssen PAEM (2004) The interaction of ocean waves and wind. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  22. 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
  23. 23.
    Ablowitz MJ, Kaup D, Newell A, Segur H (1974) The inverse scattering transform: Fourier analysis for non-linear problems. Stud Appl Math 53: 249–315MathSciNetGoogle Scholar
  24. 24.
    Newell AC (1985) Solitons in physics and mathematics. SIAMGoogle Scholar
  25. 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
  26. 26.
    Dysthe KB, Trulsen K (1999) Note on breather type solutions of the NLS as models for freak-waves. Phys Scripta T82: 48–52CrossRefADSGoogle Scholar
  27. 27.
    Ma Ya-C (1979) The perturbed plane-wave solutions of the cubic Schrödinger equation. Stud Appl Math 60: 43–58MathSciNetGoogle Scholar
  28. 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
  29. 29.
    Akhmediev NN, Eleonskii VM, Kulagin NE (1987) Exact first-order solutions of the nonlinear Schrödinger equation. Theor Math Phys 72: 809–818zbMATHCrossRefMathSciNetGoogle Scholar
  30. 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. 31.
    Dold JW, Peregrine DH (1986) Water-wave modulation. 20th international conference on coastal engineering, Taipei, vol 1. pp 163–175Google Scholar
  32. 32.
    Henderson KL, Peregrine DH, Dold JW (1999) Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrödinger equation. Wave Motion 29: 341–361zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Clamond D, Francius M, Grue J, Kharif C (2006) Long time interaction of envelope solitons and freak wave formations. Eur J Mech B Fluids 25: 536–553zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Shemer L, Goulitski K, Kit E (2007) Evolution of wide-spectrum unidirectional wave groups in a tank: an experimental and numerical study. Eur J Mech B/Fluids 26: 193–219zbMATHCrossRefGoogle Scholar
  35. 35.
    Landau LD, Lifshitz EM (1998) Quantum mechanics. Butterworth-Heinemannn, OxfordGoogle Scholar
  36. 36.
    Akhmediev N, Soto-Crespo JM, Ankiewicz A (2009) Extreme waves that appear from nowhere: on the nature of rogue waves. Phys Lett A 373: 2137–2145CrossRefMathSciNetADSGoogle Scholar
  37. 37.
    Dyachenko AI, Zakharov VE (2008) On the formation of freak waves on the surface of deep water. JETP Lett 88: 307–311CrossRefADSGoogle Scholar
  38. 38.
    Onorato M, Osborne AR, Serio M (2004) Observation of strongly non-Gaussian statistics for random sea surface gravity waves in wave flume experiments. Phys Rev E 70(6): 067302CrossRefADSGoogle Scholar
  39. 39.
    Lavrenov IV (1998) The wave energy concentration at the Agulhas current off South Africa. Nat Hazards 17: 117–127CrossRefGoogle Scholar
  40. 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
  41. 41.
    Annenkov S, Shrira V (2009) ‘Fast’ nonlinear evolution in wave turbulence. Phys Rev Lett 102: 024502CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of Mathematics, EPSAMKeele UniversityKeeleUK
  2. 2.Shirshov Institute for OceanologyMoscowRussia

Personalised recommendations