Abstract
We discuss physical and statistical properties of rogue wave generation in deep water from the perspective of the focusing Nonlinear Schrödinger equation and some of its higher order generalizations. Numerical investigations and analytical arguments based on the inverse spectral theory of the underlying integrable model, perturbation analysis, and statistical methods provide a coherent picture of rogue waves associated with nonlinear focusing events. Homoclinic orbits of unstable solutions of the underlying integrable model are certainly candidates for extreme waves, however, for more realistic models such as the modified Dysthe equation two novel features emerge: (a) a chaotic sea state appears to be an important mechanism for both generation and increased likelihood of rogue waves; (b) the extreme waves intermittently emerging from the chaotic background can be correlated with the homoclinic orbits characterized by maximal coalescence of their spatial modes.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Ablowitz MJ, Hammack J, Henderson D, Schober CM (2000) Modulated periodic stokes waves in deep water. Phys Rev Lett 84:887–890
Ablowitz MJ, Hammack J, Henderson D, Schober CM (2001) Long time dynamics of the modulational instability of deep water waves. Physica D 152–153:416–433
Ablowitz MJ, Segur H (1981) Solitons and the inverse scattering transform. SIAM, Philadelphia
Akhmediev NN, Korneev VI, Mitskevich NV (1988) N-modulation signals in a single-mode optical waveguide under nonlinear conditions. Sov Phys JETP 67:1
Bridges TJ, Derks G (1999) Unstable eigenvalues and the linearization about solitary waves and fronts with symmetry. Proc R Soc Lond A 455:2427
Cai D, McLaughlin DW, McLaughlin KTR (1995) The nonlinear Schrödinger equation as both a PDE and a dynamical system. Preprint.
Calini A, Schober CM (2002) Homoclinic chaos increases the likelihood of rogue waves. Phys Lett A 298:335–349
Calini A, Ercolani NM, McLaughlin DW, Schober CM (1996) Mel’nikov analysis of numerically induced chaos in the nonlinear Schrödinger equation. Physica D 89:227–260
Ercolani N, Forest MG, McLaughlin DW (1990) Geometry of the modulational instability. Part III: Homoclinic orbits for the periodic Sine-Gordon equation. Physica D 43:349–384
Haller G, Wiggins S (1992) Orbits homoclinic to resonances: The Hamiltonian case. Physica D 66:298–346
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
Islas A, Schober CM (2005) Predicting rogue waves in random oceanic sea states. Phys Fluids 17:1–4
Its AR, Rybin AV, Salle MA (1988) On the exact integration of the nonlinear Schrodinger equation. Theoret. and Math. Phys. 74(1): 20–32
Janssen P (2003) Nonlinear four-wave interactions and freak waves. J Phys Oceanogr 33:863–884
Karjanto N (2006) Mathematical aspects of extreme water waves. Ph.D. Thesis, Universiteet Twente
Li Y (1999) Homoclinic tubes in the nonlinear Schrödinger equation under Hamiltonian perturbations. Prog Theor Phys 101:559–577
Li Y, McLaughlin DW (1994) Morse and Mel’nikov functions for NLS Pde’s discretized perturbed NLS systems. I. Homoclinic orbits. Commun Math Phys 612:175–214
Li Y, McLaughlin DW, Shatah J, Wiggins S (1996) Persistent homoclinic orbits for a perturbed nonlinear Schrödinger equation. Commun Pure Appl Math 49:1175–1255
Longuet-Higgins MS (1952) On the statistical distribution of the heights of sea waves. J Mar Res 11:1245
Matveev VB, Salle MA (1991) Darboux transformations and solitons. Springer, Berlin Heidelberg New York
McLaughlin DW, Schober CM (1992) Chaotic and homoclinic behavior for numerical discretizations of the nonlinear Schrödinger equation. Physica D 57:447–465
Ochi MK (1998) Ocean waves: The stochastic approach. Cambridge University Press, Cambridge
Osborne A, Onorato M, Serio M (2000) The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains. Phys Lett A 275:386
Onorato M, Osborne A, Serio M, Bertone S (2001) Freak wave in random oceanic sea states. Phys Rev Lett 86:5831
Schober C (2006) Melnikov analysis and inverse spectral analysis of rogue waves in deep water. Eur J Mech B Fluids 25:602–620
Trulsen K, Dysthe K (1996) A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water. Wave Motion 24:281
Trulsen K, Dysthe K (1997a) Frequency downshift in three-dimensional wave trains in a deep basin. J Fluid Mech 352:359–373
Trulsen K, Dysthe K (1997b) Freak waves – a three dimensional wave simulation. In: Rood EP (ed) Naval hydrodynamics. Proceedings of the 21st symposium on nature. Academic Press, USA
Zakharov VE, Shabat AB (1972) Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov Phys JETP 34:62–69
Zeng C (2000) Homoclinic orbits for a perturbed nonlinear Schrödinger equation. Commun Pure Appl Math 53:1222–1283
Zeng C (2000) Erratum: Homoclinic orbits for a perturbed nonlinear Schrödinger equation. Commun Pure Appl Math 53:1603–1605
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer Science + Business Media B.V.
About this chapter
Cite this chapter
Calini, A., Schober, C.M. (2008). Rogue Waves in Higher Order Nonlinear Schrödinger Models. In: Pelinovsky, E., Kharif, C. (eds) Extreme Ocean Waves. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-8314-3_2
Download citation
DOI: https://doi.org/10.1007/978-1-4020-8314-3_2
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-8313-6
Online ISBN: 978-1-4020-8314-3
eBook Packages: Earth and Environmental ScienceEarth and Environmental Science (R0)