Rogue Waves in Higher Order Nonlinear Schrödinger Models
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.
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- Ablowitz et al.(2000)
- Ablowitz et al.(2001)
- Ablowitz and Segur(1981).Ablowitz MJ, Segur H (1981) Solitons and the inverse scattering transform. SIAM, PhiladelphiaGoogle Scholar
- Akhmediev et al.(1988)Akhmediev NN, Korneev VI, Mitskevich NV (1988) N-modulation signals in a single-mode optical waveguide under nonlinear conditions. Sov Phys JETP 67:1Google Scholar
- Bridges and Derks(1999).
- Cai et al.(1995)Cai D, McLaughlin DW, McLaughlin KTR (1995) The nonlinear Schrödinger equation as both a PDE and a dynamical system. Preprint.Google Scholar
- Calini and Schober(2002).
- Calini et al.(1996)
- Ercolani et al.(1990)
- Haller and Wiggins(1992).
- Henderson et al.(1999)
- Islas and Schober(2005).
- Its et al.(1988)
- Karjanto(2006).Karjanto N (2006) Mathematical aspects of extreme water waves. Ph.D. Thesis, Universiteet TwenteGoogle Scholar
- Li and McLaughlin(1994).
- Li et al.(1996)
- Longuet-Higgins(1952).Longuet-Higgins MS (1952) On the statistical distribution of the heights of sea waves. J Mar Res 11:1245Google Scholar
- Matveev and Salle(1991).Matveev VB, Salle MA (1991) Darboux transformations and solitons. Springer, Berlin Heidelberg New YorkGoogle Scholar
- McLaughlin and Schober(1992).
- Ochi(1998).Ochi MK (1998) Ocean waves: The stochastic approach. Cambridge University Press, CambridgeGoogle Scholar
- Trulsen and Dysthe(1996).
- Trulsen and Dysthe(1997a).
- Trulsen and Dysthe(1997a).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, USAGoogle Scholar
- Zakharov and Shabat(1972).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 34:62–69Google Scholar