Abstract

Nondeterministic giant waves, denoted as rogue, killer, monster, or freak waves, have been reported in many different branches of physics. Their physical interpretation is however still debated: despite massive numerical and experimental evidence, a solid explanation for their spontaneous formation has not been identified yet. Here we propose that rogue waves [more precisely, rogue solitons (RSs)] in optical fibers may actually result from a complex dynamical process very similar to well-known mechanisms such as glass transitions and protein folding. We describe how the interaction among optical solitons produces an energy landscape in a highly dimensional parameter space with multiple quasi-equilibrium points. These configurations have the same statistical distribution of the observed rogue events and are explored during the light dynamics due to soliton collisions, with inelastic mechanisms enhancing the process. Slightly different initial conditions lead to very different dynamics in this complex geometry; a RS turns out to stem from one particularly deep quasi-equilibrium point of the energy landscape in which the system may be transiently trapped during evolution. This explanation will prove to be fruitful to the vast community interested in freak waves.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
  50. J. Garnier, A. Picozzi, “Unified kinetic formulation of incoherent waves propagating in nonlinear media with noninstantaneous response,” Phys. Rev. A 81, 033831 (2010).
    [Crossref]
  51. M. Ruderman, “Freak waves in laboratory and space plasmas,” Eur. Phys. J. Spec. Top. 185, 57–66 (2010).
    [Crossref]
  52. L. Stenflo, M. Marklund, “Rogue waves in the atmosphere,” J. Plasma Phys. 76, 293–295 (2010).
    [Crossref]
  53. M. Onorato, D. Proment, A. Toffoli, “Freak waves in crossing seas,” Eur. Phys. J. Spec. Top. 185, 45–55 (2010).
    [Crossref]
  54. F. T. Arecchi, U. Bortolozzo, A. Montina, S. Residori, “Granularity and inhomogeneity are the joint generators of optical rogue waves,” Phys. Rev. Lett. 106, 153901 (2011).
    [Crossref]
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    [Crossref]
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  62. V. Karpman, V. Solov’ev, “A perturbation theory for soliton systems,” Phys. D 3, 142–164 (1981).
    [Crossref]
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    [Crossref]
  65. V. Gerdjikov, B. Baizakov, M. Salerno, N. Kostov, “Adiabatic N-soliton interactions of Bose–Einstein condensates in external potentials,” Phys. Rev. E 73, 046606 (2006).
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    [Crossref]
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    [Crossref]

2015 (3)

P. Walczak, S. Randoux, P. Suret, “Optical rogue waves in integrable turbulence,” Phys. Rev. Lett. 114, 143903 (2015).

G. Weerasekara, A. Tokunaga, H. Terauchi, M. Eberhard, A. Maruta, “Soliton’s eigenvalue based analysis on the generation mechanism of rogue wave phenomenon in optical fibers exhibiting weak third order dispersion,” Opt. Express 23, 143–153 (2015).
[Crossref]

C. Liu, R. E. C. van der Wel, N. Rotenberg, L. Kuipers, T. F. Krauss, A. Di Falco, A. Fratalocchi, “Triggering extreme events at the nanoscale in photonic seas,” Nat. Phys. 11, 358–363 (2015).
[Crossref]

2014 (3)

J. M. Dudley, F. Dias, M. Erkintalo, G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8, 755–764 (2014).
[Crossref]

A. Niang, F. Amrani, M. Salhi, P. Grelu, F. Sanchez, “Rains of solitons in a figure-of-eight passively mode-locked fiber laser,” Appl. Phys. B 116, 771–775 (2014).
[Crossref]

N. Mosavi, C. Nelson, B. S. Marks, B. G. Boone, C. R. Menyuk, “Aberrated beam propagation through turbulence and comparison of Monte Carlo simulations to field test measurements,” Opt. Eng. 53, 086108 (2014).
[Crossref]

2013 (8)

R. Driben, B. A. Malomed, A. V. Yulin, D. V. Skryabin, “Newton’s cradles in optics: from N-soliton fission to soliton chains,” Phys. Rev. A 87, 1–8 (2013).
[Crossref]

B. Fischer, A. Bekker, “Many-body photonics,” Opt. Photon. News 24(9), 40–47 (2013).
[Crossref]

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, F. T. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528, 47–89 (2013).
[Crossref]

A. Ankiewicz, J. M. Soto-Crespo, M. A. Chowdhury, N. N. Akhmediev, “Rogue waves in optical fibers in presence of third-order dispersion, self-steepening, and self-frequency shift,” J. Opt. Soc. Am. B 30, 87–94 (2013).
[Crossref]

U. Bandelow, N. N. Akhmediev, “Solitons on a background, rogue waves, and classical soliton solutions of the Sasa–Satsuma equation,” J. Opt. 15, 064006 (2013).
[Crossref]

P. T. S. DeVore, D. R. Solli, D. Borlaug, C. Ropers, B. Jalali, “Rogue events and noise shaping in nonlinear silicon photonics,” J. Opt. 15, 064001 (2013).
[Crossref]

N. N. Akhmediev, J. M. Dudley, D. R. Solli, S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15, 060201 (2013).
[Crossref]

G.-L. Oppo, A. M. Yao, D. Cuozzo, “Self-organization, pattern formation, cavity solitons, and rogue waves in singly resonant optical parametric oscillators,” Phys. Rev. A 88, 043813 (2013).
[Crossref]

2012 (4)

S. T. Sørensen, O. Bang, B. Wetzel, J. M. Dudley, “Describing supercontinuum noise and rogue wave statistics using higher-order moments,” Opt. Commun. 285, 2451–2455 (2012).
[Crossref]

F. Baronio, A. Degasperis, M. Conforti, S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[Crossref]

P. Grelu, N. N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6, 84–92 (2012).
[Crossref]

J. E. Prilepsky, S. A. Derevyanko, S. K. Turitsyn, “Temporal solitonic crystals and non-Hermitian informational lattices,” Phys. Rev. Lett. 108, 183902 (2012).
[Crossref]

2011 (4)

C. Conti, L. Leuzzi, “Complexity of waves in nonlinear disordered media,” Phys. Rev. B 83, 134204 (2011).
[Crossref]

F. T. Arecchi, U. Bortolozzo, A. Montina, S. Residori, “Granularity and inhomogeneity are the joint generators of optical rogue waves,” Phys. Rev. Lett. 106, 153901 (2011).
[Crossref]

J. M. Soto-Crespo, P. Grelu, N. N. Akhmediev, “Dissipative rogue waves: extreme pulses generated by passively mode-locked lasers,” Phys. Rev. E 84, 016604 (2011).
[Crossref]

S. Vergeles, S. K. Turitsyn, “Optical rogue waves in telecommunication data streams,” Phys. Rev. A 83, 061801 (2011).
[Crossref]

2010 (11)

G. Genty, C. de Sterke, O. Bang, F. Dias, N. N. Akhmediev, J. M. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phys. Lett. A 374, 989–996 (2010).
[Crossref]

B. Jalali, D. R. Solli, K. Goda, K. Tsia, C. Ropers, “Real-time measurements, rare events and photon economics,” Eur. Phys. J. Spec. Top. 185, 145–157 (2010).
[Crossref]

D. R. Solli, C. Ropers, B. Jalali, “Rare frustration of optical supercontinuum generation,” Appl. Phys. Lett. 96, 151108 (2010).
[Crossref]

Y. Zhen-Ya, “Financial rogue waves,” Commun. Theor. Phys. 947, 6–9 (2010).

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. N. Akhmediev, J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[Crossref]

M. Erkintalo, G. Genty, J. M. Dudley, “On the statistical interpretation of optical rogue waves,” Eur. Phys. J. Spec. Top. 185, 135–144 (2010).
[Crossref]

T. X. Tran, A. Podlipensky, P. St. J. Russell, F. Biancalana, “Theory of Raman multipeak states in solid-core photonic crystal fibers,” J. Opt. Soc. Am. B 27, 1785–1791 (2010).
[Crossref]

J. Garnier, A. Picozzi, “Unified kinetic formulation of incoherent waves propagating in nonlinear media with noninstantaneous response,” Phys. Rev. A 81, 033831 (2010).
[Crossref]

M. Ruderman, “Freak waves in laboratory and space plasmas,” Eur. Phys. J. Spec. Top. 185, 57–66 (2010).
[Crossref]

L. Stenflo, M. Marklund, “Rogue waves in the atmosphere,” J. Plasma Phys. 76, 293–295 (2010).
[Crossref]

M. Onorato, D. Proment, A. Toffoli, “Freak waves in crossing seas,” Eur. Phys. J. Spec. Top. 185, 45–55 (2010).
[Crossref]

2009 (7)

Y. Bludov, V. Konotop, N. N. Akhmediev, “Matter rogue waves,” Phys. Rev. A 80, 033610 (2009).
[Crossref]

L. Leuzzi, C. Conti, V. Folli, L. Angelani, G. Ruocco, “Phase diagram and complexity of mode-locked lasers: from order to disorder,” Phys. Rev. Lett. 102, 1–4 (2009).
[Crossref]

S. Chouli, P. Grelu, “Rains of solitons in a fiber laser,” Opt. Express 17, 11776–11781 (2009).
[Crossref]

A. Mussot, A. Kudlinski, M. Kolobov, E. Louvergneaux, M. Douay, M. Taki, “Observation of extreme temporal events in CW-pumped supercontinuum,” Opt. Express 17, 17010–17015 (2009).
[Crossref]

C. Garett, J. Gemmrich, “Rogue waves,” Phys. Today 62(6), 62–63 (2009).
[Crossref]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, N. N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express 17, 21497–21508 (2009).
[Crossref]

N. N. Akhmediev, J. Soto-Crespo, A. Ankiewicz, “Extreme waves that appear from nowhere: on the nature of rogue waves,” Phys. Lett. A 373, 2137–2145 (2009).
[Crossref]

2008 (2)

J. M. Dudley, G. Genty, B. J. Eggleton, “Harnessing and control of optical rogue waves in supercontinuum generation,” Opt. Express 16, 3644–3651 (2008).
[Crossref]

K. Petrowski, D. Limsui, C. Menyuk, R. Joseph, M. Thomas, W. Torruellas, “Turbulent thermal blooming,” Proc. SPIE 6951, 695104 (2008).

2007 (1)

D. R. Solli, C. Ropers, P. Koonath, B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[Crossref]

2006 (1)

V. Gerdjikov, B. Baizakov, M. Salerno, N. Kostov, “Adiabatic N-soliton interactions of Bose–Einstein condensates in external potentials,” Phys. Rev. E 73, 046606 (2006).
[Crossref]

2005 (2)

C. Conti, “Complex light: dynamic phase transitions of a light beam in a nonlinear nonlocal disordered medium,” Phys. Rev. E 72, 066620 (2005).
[Crossref]

O. Gat, A. Gordon, B. Fischer, “Light-mode locking: a new class of solvable statistical physics systems,” New J. Phys. 7, 1–11 (2005).
[Crossref]

2004 (1)

M. Hopkin, “Sea snapshots will map frequency of freak waves,” Nature 430, 492 (2004).
[Crossref]

2003 (1)

A. Gordon, B. Fischer, “Phase transition theory of pulse formation in passively mode-locked lasers with dispersion and Kerr nonlinearity,” Opt. Commun. 223, 151–156 (2003).
[Crossref]

2001 (3)

B. Rumpf, A. Newell, “Coherent structures and entropy in constrained, modulationally unstable, nonintegrable systems,” Phys. Rev. Lett. 87, 054102 (2001).
[Crossref]

P. G. Debenedetti, F. H. Stillinger, “Supercooled liquids and the glass transition,” Nature 410, 259–267 (2001).
[Crossref]

V. Gerdjikov, I. Uzunov, “Adiabatic and non-adiabatic soliton interactions in nonlinear optics,” Phys. D 152–153, 355–362 (2001).
[Crossref]

1998 (1)

S. Sastry, P. G. Debenedetti, F. H. Stillinger, “Signatures of distinct dynamical regimes in the energy landscape of a glass-forming liquid,” Nature 393, 554–557 (1998).
[Crossref]

1997 (1)

A. Schwache, F. Mitschke, “Properties of an optical soliton gas,” Phys. Rev. E 55, 7720–7725 (1997).
[Crossref]

1996 (2)

I. Uzunov, V. Gerdjikov, M. Gölles, F. Lederer, “On the description of N-soliton interaction in optical fibers,” Opt. Commun. 125, 237–242 (1996).

V. Gerdjikov, D. Kaup, I. Uzunov, E. Evstatiev, “Asymptotic behavior of N-soliton trains of the nonlinear Schrödinger equation,” Phys. Rev. Lett. 77, 3943–3946 (1996).
[Crossref]

1995 (1)

B. A. Malomed, “Bound states in a gas of solitons supported by a randomly fluctuating force,” Europhys. Lett. 30, 507–512 (1995).
[Crossref]

1993 (1)

J. Arnold, “Soliton pulse-position modulation,” IEE Proc. J. 140, 359–366 (1993).
[Crossref]

1989 (1)

C. R. Menyuk, “Transient solitons in stimulated Raman scattering,” Phys. Rev. Lett. 62, 2937–2940 (1989).
[Crossref]

1986 (2)

N. N. Akhmediev, V. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrödinger equation,” Theor. Math. Phys. 69, 1089–1093 (1986).
[Crossref]

D. Anderson, M. Lisak, “Bandwidth limits due to mutual pulse interaction in optical soliton communication systems,” Opt. Lett. 11, 174–176 (1986).
[Crossref]

1985 (1)

D. Anderson, M. Lisak, “Bandwidth limits due to incoherent soliton interaction in optical-fiber communication systems,” Phys. Rev. A 32, 2270–2274 (1985).
[Crossref]

1984 (1)

F. H. Stillinger, T. A. Weber, “Packing structures and transitions in liquids and solids,” Science 225, 983–989 (1984).
[Crossref]

1981 (1)

V. Karpman, V. Solov’ev, “A perturbation theory for soliton systems,” Phys. D 3, 142–164 (1981).
[Crossref]

1971 (1)

V. E. Zakharov, “Kinetic equation for solitons,” Sov. Phys. JETP 33, 538–541 (1971).

1970 (1)

G. M. Zaslavskii, N. N. Filonenko, “Kinetics of nonlinear waves in dispersive media,” Sov. J. Exp. Theor. Phys. 30, 676–681 (1970).

1969 (1)

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[Crossref]

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N. Mosavi, C. Nelson, B. S. Marks, B. G. Boone, C. R. Menyuk, “Aberrated beam propagation through turbulence and comparison of Monte Carlo simulations to field test measurements,” Opt. Eng. 53, 086108 (2014).
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A. Niang, F. Amrani, M. Salhi, P. Grelu, F. Sanchez, “Rains of solitons in a figure-of-eight passively mode-locked fiber laser,” Appl. Phys. B 116, 771–775 (2014).
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P. Walczak, S. Randoux, P. Suret, “Optical rogue waves in integrable turbulence,” Phys. Rev. Lett. 114, 143903 (2015).

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C. Liu, R. E. C. van der Wel, N. Rotenberg, L. Kuipers, T. F. Krauss, A. Di Falco, A. Fratalocchi, “Triggering extreme events at the nanoscale in photonic seas,” Nat. Phys. 11, 358–363 (2015).
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M. Ruderman, “Freak waves in laboratory and space plasmas,” Eur. Phys. J. Spec. Top. 185, 57–66 (2010).
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B. Rumpf, A. Newell, “Coherent structures and entropy in constrained, modulationally unstable, nonintegrable systems,” Phys. Rev. Lett. 87, 054102 (2001).
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A. Niang, F. Amrani, M. Salhi, P. Grelu, F. Sanchez, “Rains of solitons in a figure-of-eight passively mode-locked fiber laser,” Appl. Phys. B 116, 771–775 (2014).
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D. R. Solli, C. Ropers, B. Jalali, “Rare frustration of optical supercontinuum generation,” Appl. Phys. Lett. 96, 151108 (2010).
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B. Jalali, D. R. Solli, K. Goda, K. Tsia, C. Ropers, “Real-time measurements, rare events and photon economics,” Eur. Phys. J. Spec. Top. 185, 145–157 (2010).
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K. Petrowski, D. Limsui, C. Menyuk, R. Joseph, M. Thomas, W. Torruellas, “Turbulent thermal blooming,” Proc. SPIE 6951, 695104 (2008).

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M. Onorato, D. Proment, A. Toffoli, “Freak waves in crossing seas,” Eur. Phys. J. Spec. Top. 185, 45–55 (2010).
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C. Liu, R. E. C. van der Wel, N. Rotenberg, L. Kuipers, T. F. Krauss, A. Di Falco, A. Fratalocchi, “Triggering extreme events at the nanoscale in photonic seas,” Nat. Phys. 11, 358–363 (2015).
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S. Vergeles, S. K. Turitsyn, “Optical rogue waves in telecommunication data streams,” Phys. Rev. A 83, 061801 (2011).
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F. Baronio, A. Degasperis, M. Conforti, S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
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P. Walczak, S. Randoux, P. Suret, “Optical rogue waves in integrable turbulence,” Phys. Rev. Lett. 114, 143903 (2015).

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F. H. Stillinger, T. A. Weber, “Packing structures and transitions in liquids and solids,” Science 225, 983–989 (1984).
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C. Liu, R. E. C. van der Wel, N. Rotenberg, L. Kuipers, T. F. Krauss, A. Di Falco, A. Fratalocchi, “Triggering extreme events at the nanoscale in photonic seas,” Nat. Phys. 11, 358–363 (2015).
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Nature (4)

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Opt. Eng. (1)

N. Mosavi, C. Nelson, B. S. Marks, B. G. Boone, C. R. Menyuk, “Aberrated beam propagation through turbulence and comparison of Monte Carlo simulations to field test measurements,” Opt. Eng. 53, 086108 (2014).
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Phys. Rep. (1)

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, F. T. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528, 47–89 (2013).
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Y. Bludov, V. Konotop, N. N. Akhmediev, “Matter rogue waves,” Phys. Rev. A 80, 033610 (2009).
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J. Garnier, A. Picozzi, “Unified kinetic formulation of incoherent waves propagating in nonlinear media with noninstantaneous response,” Phys. Rev. A 81, 033831 (2010).
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S. Vergeles, S. K. Turitsyn, “Optical rogue waves in telecommunication data streams,” Phys. Rev. A 83, 061801 (2011).
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G.-L. Oppo, A. M. Yao, D. Cuozzo, “Self-organization, pattern formation, cavity solitons, and rogue waves in singly resonant optical parametric oscillators,” Phys. Rev. A 88, 043813 (2013).
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P. Walczak, S. Randoux, P. Suret, “Optical rogue waves in integrable turbulence,” Phys. Rev. Lett. 114, 143903 (2015).

B. Rumpf, A. Newell, “Coherent structures and entropy in constrained, modulationally unstable, nonintegrable systems,” Phys. Rev. Lett. 87, 054102 (2001).
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J. E. Prilepsky, S. A. Derevyanko, S. K. Turitsyn, “Temporal solitonic crystals and non-Hermitian informational lattices,” Phys. Rev. Lett. 108, 183902 (2012).
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L. Leuzzi, C. Conti, V. Folli, L. Angelani, G. Ruocco, “Phase diagram and complexity of mode-locked lasers: from order to disorder,” Phys. Rev. Lett. 102, 1–4 (2009).
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V. Gerdjikov, D. Kaup, I. Uzunov, E. Evstatiev, “Asymptotic behavior of N-soliton trains of the nonlinear Schrödinger equation,” Phys. Rev. Lett. 77, 3943–3946 (1996).
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F. T. Arecchi, U. Bortolozzo, A. Montina, S. Residori, “Granularity and inhomogeneity are the joint generators of optical rogue waves,” Phys. Rev. Lett. 106, 153901 (2011).
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K. Petrowski, D. Limsui, C. Menyuk, R. Joseph, M. Thomas, W. Torruellas, “Turbulent thermal blooming,” Proc. SPIE 6951, 695104 (2008).

Science (1)

F. H. Stillinger, T. A. Weber, “Packing structures and transitions in liquids and solids,” Science 225, 983–989 (1984).
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G. M. Zaslavskii, N. N. Filonenko, “Kinetics of nonlinear waves in dispersive media,” Sov. J. Exp. Theor. Phys. 30, 676–681 (1970).

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Supplementary Material (1)

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Figures (5)

Fig. 1.
Fig. 1.

Potential energy landscape and the emergence of RSs in optical fibers. (a,b) A pictorial representation of the energy landscape of a complex system represented as a two-dimensional surface: we observe many local minima with various depths and locations, separated by saddle points. The dynamics is strongly influenced by the initial conditions: we highlight two different trajectories, one dropping into a deep minimum of the potential energy (white solid line) and one experiencing many shallow minima (yellow solid line). This explains why slightly different noise configurations around the input pulse generate completely different dynamics, leading only rarely to RSs. In panel (c), we compare this simplified image with the interaction of many solitons. It is characterized by evolution in a multidimensional complex topology with many minima and saddles. Rare events, due to collisions or higher-order effects (such as dispersion and Raman gain), trigger the system in regions o lower energy, which is in our case the Hamiltonian of the NLS. We represent the energy landscape generated by soliton interaction by different values of H at different numerically computed equilibria. The horizontal axis enumerates the solutions obtained from different initial random conditions. The solution marked by the filled circle is the exceptional solution, which we show in Fig. 2.

Fig. 2.
Fig. 2.

Equilibria of the weak interaction of the NLS solitons ( N = 40 ) and their statistical properties. (a and b) Characterization of an exceptional solution with an anomalously large peak: (a) a fixed point of the WIM, sorted according to position ξ k ; (b) squared amplitude and phase of the reconstructed field (the ellipses and arrows are meant to guide to the corresponding axis), which shows a three-peak structure and an apparent sign change across the pulse (the phase exhibits two subsequent jumps); in the inset, we compare the structure factor of the present solution (blue solid line) with a crystal, the lattice constant of which is Δ ξ ave (green dashed line). In contrast to (b), we show in (c) two different equilibrium solutions of the WIM, which are much less intense and present a multiple-peak structure: in (c1), the peak is one order of magnitude smaller than that in panel (b) and a trailing pulse appears, whereas (c2) exhibits two peaks. These solutions correspond to the shallower minima in Fig. 1(c). Finally, in (d), the statistical analysis of the ensemble of WIM solutions is presented: a histogram of amplitudes exceeding the Q = 0.95 quantile of the distribution (normalized such that the total area of the bars sums up to unity). In the inset, the Weibull fit of the distribution is shown in double logarithmic scale ( N = 40 ).

Fig. 3.
Fig. 3.

Evolution of the temporal profile along the propagation direction: at Z = 25 m , (a) corresponds to the maximum peak (a type-I RS) and (b) corresponds to the most redshifted output (a type-II RS). In (a), we observe a sequence of collisions ending in an event at the fiber output. In (b), we identify the collision event that leads to the emergence of the most redshifted soliton.

Fig. 4.
Fig. 4.

Temporal profile at collision points. The time variable is centered and scaled by T MI and we included events occurring after Z = 25 m . Panels (a–c) show three collision events for the type-I event: while a collision occurs near Z 18 m [see Fig. 3(a)], a major event occurs at Z 25 m . In (d–f), we consider a type-II scenario: in this case, the collision at Z 18 m is the main event and the amplitude peak is much stronger than that at any other propagation stage. The amplitude and phase waveform are in qualitative agreement between the analytical model and the simulation results [see Figs. 2(b) and 2(c)].

Fig. 5.
Fig. 5.

Hamiltonian evolution H ( z ) . We separate (a) the kinetic (linear) part H K and (b) the nonlinear part H NL . We report the two scenarios described in the text and shown in Fig. 3: the blue solid line corresponds to a type-I RS generation, whereas the red dashed line corresponds to the type-II mechanism. In order to thoroughly characterize the evolution of the two scenarios, we show the evolution up to L = 40 m and normalize these quantities in order to compare H NL with the energy landscape in Fig. 1(b) (see the main text). We notice that H NL exhibits many local minima, which correspond to step increases in H K and the latter tends to settle on a linear growth. Importantly, each minimum corresponds to a soliton collision. The arrows highlight the main events at Z 18 m and Z 25 m , for the two scenarios. The minima correspond to an increase in H K , which then settles on a steady linear growth. The deeper minimum in the type-II scenario leads to a sudden increase to larger values of H K , whereas type-I is characterized by smaller steps.

Equations (5)

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i u z + 1 2 2 u t 2 + | u | 2 u = 0 ,
u ( z , t ) = k = 1 N u k ( z , t ) ,
u k ( z , t ) = 2 ν k sech [ 2 ν k ( t ξ k ) ] e i 2 μ k ( t ξ k ) + i δ k ,
H ( z ) = ( z , t ) d t ,
i U Z + k 2 i k k ! β k k U T k + γ ( 1 + i τ shock T ) U 0 R ( T ) | U ( T T ) | 2 d T = 0 ,

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