Skip to main content
SpringerLink
Log in

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

  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Benjamin TB, Feir JE (1967) The disintegration of wavetrains in deep water. Part 1. J Fluid Mech 27: 417–430

    Article  MATH  ADS  Google Scholar 

  2. Benney DJ, Roskes GJ (1969) Wave instabilities. Stud Appl Math 48: 377–385

    MATH  Google Scholar 

  3. Peregrine DH (1983) Water waves, nonlinear Schrödinger equations and their solutions. J Aust Math Soc Ser B 25: 16–43

    Article  MATH  MathSciNet  Google Scholar 

  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: 386–393

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Dyachenko AI, Zakharov VE (2005) Modulational instability of Stokes waves → freak wave. JETP Lett 81: 255–259

    Article  ADS  Google Scholar 

  6. Kharif C, Pelinovsky E (2003) Physical mechanisms of the rogue wave phenomenon. Eur J Fluid Mech B/Fluids 22: 603–634

    Article  MATH  MathSciNet  ADS  Google Scholar 

  7. Kharif C, Pelinovsky E, Slyunyaev A (2009) Rogue waves in the ocean. Springer, Berlin

    Google Scholar 

  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: 2541–2558

    MathSciNet  Google Scholar 

  9. Tanaka M (1990) Maximum amplitude of modulated wave train. Wave Motion 12: 559–568

    Article  Google Scholar 

  10. Alber IE (1978) The Effects of Randomness on the Stability of Two-Dimensional Surface Wave trains. Proc R Soc Lond A 363(1715): 525–546

    Article  MATH  MathSciNet  ADS  Google Scholar 

  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

    Article  MATH  MathSciNet  ADS  Google Scholar 

  12. Stiassnie M, Regev A, Agnon Y (2008) Recurrent solutions of Alber’s equation for random water-wave fields. J Fluid Mech 598: 245–266

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. Janssen PAEM (2003) Nonlinear four-wave interactions and freak waves. J Phys Oceanogr 33: 863–884

    Article  MathSciNet  ADS  Google Scholar 

  14. Annenkov S, Shrira V (2009) Evolution of kurtosis for wind waves. Geophys Res Lett 36: L13603. doi:10.1029/2009GL038613

    Article  ADS  Google 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 16–43

  16. Islas LA, Schober CM (2005) Predicting rogue waves in random oceanic sea states. Phys Fluids 17: 0317011-4

    Article  MathSciNet  Google Scholar 

  17. Slunyaev A, Pelinovsky E, Guedes Soares C (2005) Modelling freak waves from the North Sea. Appl Ocean Res 27: 12–22

    Article  Google Scholar 

  18. Slunyaev A (2006) Nonlinear analysis and simulations of measured freak wave time series. Eur J Mech -B Fluids 25: 621–635

    Article  MATH  MathSciNet  Google Scholar 

  19. Akhmediev N, Ankiewicz A, Taki M (2008) Waves that appear from nowhere and disappear without a trace. Phys Lett A 373: 675–678

    Article  ADS  Google Scholar 

  20. Mei CC, Stiassnie M, Yue DK-P (2005) Theory and applications of ocean surface waves. World Sci

  21. Janssen PAEM (2004) The interaction of ocean waves and wind. Cambridge University Press, Cambridge

    Book  Google 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: 118–134

    Google Scholar 

  23. Ablowitz MJ, Kaup D, Newell A, Segur H (1974) The inverse scattering transform: Fourier analysis for non-linear problems. Stud Appl Math 53: 249–315

    MathSciNet  Google Scholar 

  24. Newell AC (1985) Solitons in physics and mathematics. SIAM

  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)

  26. Dysthe KB, Trulsen K (1999) Note on breather type solutions of the NLS as models for freak-waves. Phys Scripta T82: 48–52

    Article  ADS  Google Scholar 

  27. Ma Ya-C (1979) The perturbed plane-wave solutions of the cubic Schrödinger equation. Stud Appl Math 60: 43–58

    MathSciNet  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 

  29. Akhmediev NN, Eleonskii VM, Kulagin NE (1987) Exact first-order solutions of the nonlinear Schrödinger equation. Theor Math Phys 72: 809–818

    Article  MATH  MathSciNet  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 163–175

  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: 341–361

    Article  MATH  MathSciNet  Google Scholar 

  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: 536–553

    Article  MATH  MathSciNet  Google Scholar 

  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: 193–219

    Article  MATH  Google Scholar 

  35. Landau LD, Lifshitz EM (1998) Quantum mechanics. Butterworth-Heinemannn, Oxford

    Google Scholar 

  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: 2137–2145

    Article  MathSciNet  ADS  Google Scholar 

  37. Dyachenko AI, Zakharov VE (2008) On the formation of freak waves on the surface of deep water. JETP Lett 88: 307–311

    Article  ADS  Google Scholar 

  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): 067302

    Article  ADS  Google Scholar 

  39. Lavrenov IV (1998) The wave energy concentration at the Agulhas current off South Africa. Nat Hazards 17: 117–127

    Article  Google 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): 114502

    Article  ADS  Google Scholar 

  41. Annenkov S, Shrira V (2009) ‘Fast’ nonlinear evolution in wave turbulence. Phys Rev Lett 102: 024502

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, EPSAM, Keele University, Keele, ST5 5BG, UK

    Victor I. Shrira

  2. Shirshov Institute for Oceanology, 36, Nahimovski prospect, Moscow, 117997, Russia

    Vladimir V. Geogjaev

Authors
  1. Victor I. Shrira
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vladimir V. Geogjaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Victor I. Shrira.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shrira, V.I., Geogjaev, V.V. What makes the Peregrine soliton so special as a prototype of freak waves?. J Eng Math 67, 11–22 (2010). https://doi.org/10.1007/s10665-009-9347-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10665-009-9347-2

Keywords

Search

Navigation