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2016 | OriginalPaper | Buchkapitel

Rogue Waves in Higher Order Nonlinear Schrödinger Models

verfasst von : Constance M. Schober, Annalisa Calini

Erschienen in: Extreme Ocean Waves

Verlag: Springer International Publishing

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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.

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Literatur
Zurück zum Zitat Ablowitz MJ, Segur H (1981) Solitons and the inverse scattering transform. SIAM, PhiladelphiaCrossRef Ablowitz MJ, Segur H (1981) Solitons and the inverse scattering transform. SIAM, PhiladelphiaCrossRef
Zurück zum Zitat Ablowitz MJ, Hammack J, Henderson D, Schober CM (2000) Modulated periodic stokes waves in deep water. Phys Rev Lett 84:887–890CrossRef Ablowitz MJ, Hammack J, Henderson D, Schober CM (2000) Modulated periodic stokes waves in deep water. Phys Rev Lett 84:887–890CrossRef
Zurück zum Zitat Ablowitz MJ, Hammack J, Henderson D, Schober CM (2001) Long time dynamics of the modulational instability of deep water waves. Phys D 152–153:416–433CrossRef Ablowitz MJ, Hammack J, Henderson D, Schober CM (2001) Long time dynamics of the modulational instability of deep water waves. Phys D 152–153:416–433CrossRef
Zurück zum Zitat Akhmediev NN, Korneev VI, Mitskevich NV (1988) \(N\)-modulation signals in a single-mode optical waveguide under nonlinear conditions. Sov Phys JETP 67:1 Akhmediev NN, Korneev VI, Mitskevich NV (1988) \(N\)-modulation signals in a single-mode optical waveguide under nonlinear conditions. Sov Phys JETP 67:1
Zurück zum Zitat Bridges TJ, Derks G (1999) Unstable eigenvalues and the linearization about solitary waves and fronts with symmetry. Proc R Soc Lond A 455:2427CrossRef Bridges TJ, Derks G (1999) Unstable eigenvalues and the linearization about solitary waves and fronts with symmetry. Proc R Soc Lond A 455:2427CrossRef
Zurück zum Zitat Cai D, McLaughlin DW, McLaughlin KTR (1995) The nonlinear Schrödinger equation as both a PDE and a dynamical system. Preprint Cai D, McLaughlin DW, McLaughlin KTR (1995) The nonlinear Schrödinger equation as both a PDE and a dynamical system. Preprint
Zurück zum Zitat Calini A, Schober CM (2001) Chaotic dynamics for a symmetry breaking perturbation of the NLS equation. J Math Comput Simul 55:351–364CrossRef Calini A, Schober CM (2001) Chaotic dynamics for a symmetry breaking perturbation of the NLS equation. J Math Comput Simul 55:351–364CrossRef
Zurück zum Zitat Calini A, Schober CM (2002) Homoclinic chaos increases the likelihood of rogue waves. Phys Lett A 298:335–349CrossRef Calini A, Schober CM (2002) Homoclinic chaos increases the likelihood of rogue waves. Phys Lett A 298:335–349CrossRef
Zurück zum Zitat Calini A, Ercolani NM, McLaughlin DW, Schober CM (1996) Mel’nikov analysis of numerically induced chaos in the nonlinear Schrödinger equation. Phys D 89:227–260CrossRef Calini A, Ercolani NM, McLaughlin DW, Schober CM (1996) Mel’nikov analysis of numerically induced chaos in the nonlinear Schrödinger equation. Phys D 89:227–260CrossRef
Zurück zum Zitat Dysthe K, Trulsen K (1999) Note on breather type solutions of the NLS as model for freak waves. Phys Scr T82:48–52CrossRef Dysthe K, Trulsen K (1999) Note on breather type solutions of the NLS as model for freak waves. Phys Scr T82:48–52CrossRef
Zurück zum Zitat Ercolani N, Forest MG, McLaughlin DW (1990) Geometry of the modulational instability part III: homoclinic orbits for the periodic Sine-Gordon equation. Phys D 43:349–384CrossRef Ercolani N, Forest MG, McLaughlin DW (1990) Geometry of the modulational instability part III: homoclinic orbits for the periodic Sine-Gordon equation. Phys D 43:349–384CrossRef
Zurück zum Zitat Haller G, Wiggins S (1992) Orbits homoclinic to resonances: the Hamiltonian case. Phys D 66:298–346CrossRef Haller G, Wiggins S (1992) Orbits homoclinic to resonances: the Hamiltonian case. Phys D 66:298–346CrossRef
Zurück zum Zitat Henderson KL, Peregrine DH, Dold JW (1999) Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrödinger equation. Wave Motion 29:341CrossRef Henderson KL, Peregrine DH, Dold JW (1999) Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrödinger equation. Wave Motion 29:341CrossRef
Zurück zum Zitat Islas A, Schober CM (2005) Predicting rogue waves in random oceanic sea states. Phys Fluids 17:1–4CrossRef Islas A, Schober CM (2005) Predicting rogue waves in random oceanic sea states. Phys Fluids 17:1–4CrossRef
Zurück zum Zitat Its AR, Salle MA, Rybin AV (1988) On exact integration of nonlinear Schrödinger equation. Teor Mat Fiz 74:29–45CrossRef Its AR, Salle MA, Rybin AV (1988) On exact integration of nonlinear Schrödinger equation. Teor Mat Fiz 74:29–45CrossRef
Zurück zum Zitat Janssen P (2003) Nonlinear four-wave interactions and freak waves. J Phys Oceanogr 33:863–884CrossRef Janssen P (2003) Nonlinear four-wave interactions and freak waves. J Phys Oceanogr 33:863–884CrossRef
Zurück zum Zitat Karjanto N (2006) Mathematical aspects of extreme water waves. Ph D thesis, Universiteet Twente Karjanto N (2006) Mathematical aspects of extreme water waves. Ph D thesis, Universiteet Twente
Zurück zum Zitat Kharif C, Pelinovsky E (2001) Focusing of nonlinear wave groups in deep water. JETP Lett 73:170–175 Kharif C, Pelinovsky E (2001) Focusing of nonlinear wave groups in deep water. JETP Lett 73:170–175
Zurück zum Zitat Kharif C, Pelinovsky E (2004) Physical mechanisms of the Rogue wave phenomenon. Eur J Mech B/Fluids 22:603–634CrossRef Kharif C, Pelinovsky E (2004) Physical mechanisms of the Rogue wave phenomenon. Eur J Mech B/Fluids 22:603–634CrossRef
Zurück zum Zitat Krichever IM (1977) Methods of algebraic geometry in the theory of nonlinear equations. Russ Math Surv 32:185–213CrossRef Krichever IM (1977) Methods of algebraic geometry in the theory of nonlinear equations. Russ Math Surv 32:185–213CrossRef
Zurück zum Zitat Li Y (1999) Homoclinic tubes in the nonlinear Schrödinger equation under Hamiltonian perturbations. Prog Theor Phys 101:559–577CrossRef Li Y (1999) Homoclinic tubes in the nonlinear Schrödinger equation under Hamiltonian perturbations. Prog Theor Phys 101:559–577CrossRef
Zurück zum Zitat Li Y, McLaughlin DW (1994) Morse and Mel’nikov functions for NLS PDE’s discretized perturbed NLS systems I. Homoclinic orbits. Commun Math Phys 612:175–214CrossRef Li Y, McLaughlin DW (1994) Morse and Mel’nikov functions for NLS PDE’s discretized perturbed NLS systems I. Homoclinic orbits. Commun Math Phys 612:175–214CrossRef
Zurück zum Zitat Li Y, McLaughlin DW, Shatah J, Wiggins S (1996) Persistent homoclinic orbits for a perturbed nonlinear Schrödinger equation. Commun. Pure Appl Math 49:1175–1255CrossRef Li Y, McLaughlin DW, Shatah J, Wiggins S (1996) Persistent homoclinic orbits for a perturbed nonlinear Schrödinger equation. Commun. Pure Appl Math 49:1175–1255CrossRef
Zurück zum Zitat Longuet-Higgins MS (1952) On the statistical distribution of the heights of sea waves. J Mar Res 11:1245 Longuet-Higgins MS (1952) On the statistical distribution of the heights of sea waves. J Mar Res 11:1245
Zurück zum Zitat Matveev VB, Salle MA (1991) Darboux Transformations and solitons. Springer, BerlinCrossRef Matveev VB, Salle MA (1991) Darboux Transformations and solitons. Springer, BerlinCrossRef
Zurück zum Zitat McLaughlin DW, Schober CM (1992) Chaotic and homoclinic behavior for numerical discretizations of the nonlinear Schrodinger equation. Phys D 57:447–465CrossRef McLaughlin DW, Schober CM (1992) Chaotic and homoclinic behavior for numerical discretizations of the nonlinear Schrodinger equation. Phys D 57:447–465CrossRef
Zurück zum Zitat Ochi MK (1998) Ocean waves: the stochastic approach. Cambridge University Press, CambridgeCrossRef Ochi MK (1998) Ocean waves: the stochastic approach. Cambridge University Press, CambridgeCrossRef
Zurück zum Zitat Osborne A, Onorato M, Serio M (2000) The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains. Phys Lett A 275:386CrossRef Osborne A, Onorato M, Serio M (2000) The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains. Phys Lett A 275:386CrossRef
Zurück zum Zitat Onorato M, Osborne A, Serio M, Bertone S (2001) Freak wave in random oceanic sea states. Phys Rev Lett 86:5831CrossRef Onorato M, Osborne A, Serio M, Bertone S (2001) Freak wave in random oceanic sea states. Phys Rev Lett 86:5831CrossRef
Zurück zum Zitat Schober C (2006) Melnikov analysis and inverse spectral analysis of Rogue waves in deep water. Eur J Mech B-Fluids 25:602–620CrossRef Schober C (2006) Melnikov analysis and inverse spectral analysis of Rogue waves in deep water. Eur J Mech B-Fluids 25:602–620CrossRef
Zurück zum Zitat Torcini A, Frauenkron H, Grassberger P (1997) Studies of phase turbulence in the one-dimensional complex Ginzburg-Landau equation. Phys Rev E 55:5073–5081CrossRef Torcini A, Frauenkron H, Grassberger P (1997) Studies of phase turbulence in the one-dimensional complex Ginzburg-Landau equation. Phys Rev E 55:5073–5081CrossRef
Zurück zum Zitat Trulsen K, Dysthe K (1996) A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water. Wave Motion 24:281CrossRef Trulsen K, Dysthe K (1996) A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water. Wave Motion 24:281CrossRef
Zurück zum Zitat 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 (1997a) Frequency downshift in three-dimensional wave trains in a deep basin. J Fluid Mech 352:359–373
Zurück zum Zitat Trulsen K, Dysthe K (1997b) Freak waves—a three dimensional wave simulation. In: Rood EP (ed) Proceedings of the 21st symposium naval hydrodynamics. National Academy Press Trulsen K, Dysthe K (1997b) Freak waves—a three dimensional wave simulation. In: Rood EP (ed) Proceedings of the 21st symposium naval hydrodynamics. National Academy Press
Zurück zum Zitat van Groesen EWC, Karjanto N, Peterson P, Andonowati A, Wave dislocation and nonlinear amplitude amplification for extreme fluid surface waves. Preprint van Groesen EWC, Karjanto N, Peterson P, Andonowati A, Wave dislocation and nonlinear amplitude amplification for extreme fluid surface waves. Preprint
Zurück zum Zitat White BS, Fornberg B (1998) On the chance of freak waves at sea. J Fluid Mech 355:113–138CrossRef White BS, Fornberg B (1998) On the chance of freak waves at sea. J Fluid Mech 355:113–138CrossRef
Zurück zum Zitat 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 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
Zurück zum Zitat Zeng C (2000a) Homoclinic orbits for a perturbed nonlinear Schrödinger equation. Commun Pure Appl Math 53:1222–1283 Zeng C (2000a) Homoclinic orbits for a perturbed nonlinear Schrödinger equation. Commun Pure Appl Math 53:1222–1283
Zurück zum Zitat Zeng C (2000b) Erratum: Homoclinic orbits for a perturbed nonlinear Schrödinger equation. Commun Pure Appl Math 53:1603–1605 Zeng C (2000b) Erratum: Homoclinic orbits for a perturbed nonlinear Schrödinger equation. Commun Pure Appl Math 53:1603–1605
Metadaten
Titel
Rogue Waves in Higher Order Nonlinear Schrödinger Models
verfasst von
Constance M. Schober
Annalisa Calini
Copyright-Jahr
2016
DOI
https://doi.org/10.1007/978-3-319-21575-4_1