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

Journal of Engineering Mathematics

Volumn 67, Issue 1, 2010, Pages 11-22

  Shrira, Victor I ^a   Geogjaev, Vladimir V ^b  

^a KEELE UNIVERSITY   (United Kingdom)

^b SHIRSHOV INSTITUTE OF OCEANOLOGY   (Russian Federation)

Author keywords

Breathers; Nonlinear Schr dinger equation; Pulses in optical fibres; Rogue waves; Water waves

Indexed keywords

APRIORI; BACKGROUND LEVEL; CRITICAL VALUE; CRITICALITY CONDITION; DINGER EQUATION; FREAK WAVE; INITIAL CONDITIONS; INITIAL PERTURBATION; LEADING ORDERS; NLS EQUATIONS; OPTICAL FIBRE; PEAK AMPLITUDE; PRESENCE/ABSENCE; PULSES IN OPTICAL FIBRES; REAL-WORLD; ROGUE WAVES; SMALL PERTURBATIONS; SPATIAL SCALE; SQUARE ROOTS; SUPER-CRITICAL; WAVE PATTERNS; WAVE TRAIN;

CRITICALITY (NUCLEAR FISSION); HYDRODYNAMICS; OPTICAL FIBERS; OPTICAL MATERIALS; SOLITONS; WATER WAVES;

NONLINEAR EQUATIONS;

EID: 77951975709     PISSN: 00220833     EISSN: 15732703     Source Type: Journal    
DOI: 10.1007/s10665-009-9347-2     Document Type: Article

References (41)

1. Benjamin TB, Feir JE (1967) The disintegration of wavetrains in deep water. Part 1. J Fluid Mech 27: 417-430.
2. Benney DJ, Roskes GJ (1969) Wave instabilities. Stud Appl Math 48: 377-385.
3. Peregrine DH (1983) Water waves, nonlinear Schrödinger equations and their solutions. J Aust Math Soc Ser B 25: 16-43.
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.
5. Dyachenko AI, Zakharov VE (2005) Modulational instability of Stokes waves → freak wave. JETP Lett 81: 255-259.
6. Kharif C, Pelinovsky E (2003) Physical mechanisms of the rogue wave phenomenon. Eur J Fluid Mech B/Fluids 22: 603-634.
7. Kharif C, Pelinovsky E, Slyunyaev A (2009) Rogue waves in the ocean. Springer, Berlin.
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.
9. Tanaka M (1990) Maximum amplitude of modulated wave train. Wave Motion 12: 559-568.
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.
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.
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.
13. Janssen PAEM (2003) Nonlinear four-wave interactions and freak waves. J Phys Oceanogr 33: 863-884.
14. Annenkov S, Shrira V (2009) Evolution of kurtosis for wind waves. Geophys Res Lett 36: L13603. doi: 10. 1029/2009GL038613.
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.
17. Slunyaev A, Pelinovsky E, Guedes Soares C (2005) Modelling freak waves from the North Sea. Appl Ocean Res 27: 12-22.
18. Slunyaev A (2006) Nonlinear analysis and simulations of measured freak wave time series. Eur J Mech -B Fluids 25: 621-635.
19. Akhmediev N, Ankiewicz A, Taki M (2008) Waves that appear from nowhere and disappear without a trace. Phys Lett A 373: 675-678.
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.
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.
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.
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.
27. Ma Y.-C. (1979) The perturbed plane-wave solutions of the cubic Schrödinger equation. Stud Appl Math 60: 43-58.
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).
29. Akhmediev NN, Eleonskii VM, Kulagin NE (1987) Exact first-order solutions of the nonlinear Schrödinger equation. Theor Math Phys 72: 809-818.
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).
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.
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.
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.
35. Landau LD, Lifshitz EM (1998) Quantum mechanics. Butterworth-Heinemannn, Oxford.
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.
37. Dyachenko AI, Zakharov VE (2008) On the formation of freak waves on the surface of deep water. JETP Lett 88: 307-311.
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.
39. Lavrenov IV (1998) The wave energy concentration at the Agulhas current off South Africa. Nat Hazards 17: 117-127.
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.
41. Annenkov S, Shrira V (2009) 'Fast' nonlinear evolution in wave turbulence. Phys Rev Lett 102: 024502.