Crisis at the Origin of Deterministic Rogue Waves

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Crisis at the Origin of Deterministic Rogue Waves

• PPME, Universite de la Nouvelle Caledonie
• C. Metayer, A. Serres, J. Tredicce
• INLN, UMR 6618 UNS-CNRS France
• S. Barland, M. Giudici
• CEILAP - CITEDEF Argentina
• A. Hnilo, M. Kovalski
• Univ. Politecn. Cataluna, Spain
• Masoller, C.
• Univ. Fed Pernambuco, Recife, PE Brazil
• W. Barbosa, F. Menezes DAguiar,
• J. Rios Leite, Rosero E.

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• According to fishermen tales from a pub in
  Ireland, rogue waves like solid walls of water,
  higher than 30 meters, are more or less common
  phenomena in deep ocean waters.

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Is it true? Are rogue waves so common?

• This fact is in contradiction with the Gaussian
  models used to describe fluctuations of the wave
  height in the sea.
• M. S. Longuet-Higgins, Phil. Trans. Roy. Soc. A
  249 321 (1957).
• S. Aberg and G. Lindgren, Height distribution
  of stochastic Lagrange ocean waves, Prob. Eng.
  Mech. 23, 359 (2008)
• HOWEVER

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Ferry rescue after freak wave in Irish Sea
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The freighter Riverdance was hit by a giant wave
during severe gales in the Irish Sea..
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But.What is the definition of a rogue wave?

• Old Recipe Take the 1/3 biggest amplitude waves
  calculate their average value multiply by
  2.whatever amplitude exceeds such value is a
  rogue wave!!!
• More Recent Recipe Take the probability
  distribution calculate s multiply by 4
  whatever.
• and if you want a BIG BIG rogue wavemultiply by
  8

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• In the WEBIt is probably sufficient to say that
  any wave so large that it is unexpected based on
  current conditions can be counted as a rogue.
• There are very few photographs of rogue waves.
  For centuries, the best evidence for their
  existence was anecdotal -- the countless stories
  told by sailors who had survived one.

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Some Bibliography about Rogue Waves

• Osborne, A.R. et al. Phys. Lett. A 275, 386
  (2000) and PRL 96, 014503 (2006).
• Clauss, G.F. Appl. Ocean Res. 24, 147 (2002)
  Dramas of the sea episodic waves and their
  impact on offshore structures.
• Kharif, C. and Pelinovsky E. EJ of
  Mechan.B/Fluids 22, 603 (2003).
• Petrova, P. and Guedes Soares C. Appl. Ocean
  Res. 30, 144 (2008).
• Dyachenko, A. and Zakharov, V.E. JETP lett. 81,
  255 (2005).

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How was that Opticians got interested on Rogue
Waves?

• A NONLINEAR OPTICS PHYSICIST WENT TO THE IRISH
  PUB.and then some papers appear in Nature or
  other GO..O..D Journals
• D. R. Solli, C. Ropers et al, Optical rogue
  waves, Nature 450 1054 (2007).
• B. Kibler, J. Fatome, C. Finot, G. Millot, F.
  Dias, G. Genty, N. Akhmediev and J. M. Dudley,
  The Peregrine soliton in nonlinear fibre optics
  Nature Phys. 6, 790 (2010).
• A. Montina, U. Bortolozzo, S. Residori, F.T.
  Arecchi, Phys. Rev. Lett. 103, 173901 (2009)

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Our First Experiments.

• 1) Mode Locked TiSa laser
• Hnilo et al. (Opt. Lett. November 2011)
• 2) Semiconductor Laser with Injected Signal
• Bonatto et al. (PRL, July 2011)
• 3) Laser with saturable absorber (Journal of
  Optics, submitted)

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Laser with Injected Signal
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Probability distribution of maxima
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Our Already Published Conclusions

• 1) Extreme Events are rare but they can be much
  more probable than in Gaussian models when the
  dynamical behavior is Deterministically
  Chaotic
• 2) There is chaos without rogue waves and chaos
  with rogue waves

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Some questions

• How? What is the dynamical process the laser use
  to generate extreme events?
• Can we predict deterministic extreme events in
  optical systems?
• Can we control them?

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How?

• a) Intermittency .
• P Gaspard and X Wang, PNAS 1988
• Nicolis et al., Journal of Statistical physics
  1995
• b) By abrupt expansion of a chaotic attractor??

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Bifurcation Diagrams
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Experimental results
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Laser with Modulated parameter

• Remembering very old  times 
• H.G. Solari J, E. Eschenazi, R. Gilmore et al.,
  Opt. Commun. 64, 49 (1987)
• on
• Crisis of chaotic attractors
• Two ingredients 1) chaos
• 2) Enough low dissipation in order to have
  generalized multistability (several stable
  dynamical solutions for the same parameter
  values)

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Crisis of chaotic attractors
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External crisis in a laser with mopdulated
parameter
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Then extreme events appear after an external
crisis
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Predicting Rogue waves?
In a deterministic system, the time of
prediction equals the inverse of the maximum
positive Lyapunov exponent
But in the laser with injected signal, the
prediction time is much larger, and just looking
one variable the intensity
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Conclusions

• External crisis produce abrupt expansion of
  chaotic attractors and are at the origin of some
  extreme events
• Deterministic extreme events could be predicted
  with  some  anticipation
• I still do not know if we are able to control
  deterministic extreme events
• BUT

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I am always looking for the rogue waves in New
Caledonia
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Laser with saturable absorber in Q-switch regime
(to be subm. to special issue)

• With Alejandro Hnilo and Marcelo Kovalski,
• CEILAP, Villa Martelli, Argentina

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Relevance of Spatial Effects
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Theoretical results without spatial effects
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Number of rogue waves in parameter space in LIS
(from J. Zamora)
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Some bibliography to take into account

• V. Balakrishnan, C. Nicolis, and G. Nicolis
  Extreme Value Distributions in Chaotic Dynamics
  J. of Stat.Phys. 80, 307 1995
• C. Nicolis,V. Balakrishnan, and G. Nicolis
  Extreme Events in Deterministic Dynamical
  Systems PRL 97, 210602 (2006)
• P. Gaspard and X.J. Wang Sporadicity between
  periodic and chaotic dynamical behaviors Proc.
  Nat. Acad. Sci. USA 85, 4591 (1988).

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Perspectives

• 1) Experiment of laser with modulation in solid
  state laser (at CEILAP). Why solid state and not
  semiconductor at INLN?
• 2) Experiments laser with injection large Fresnel
  number (if INLN agree)
• 3) large fresnel number edge emitter lasers
  (UFPE)
• 4) laser with feedback (UPC) theory
• 5) Numerical work at UNC

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Conclusions

• Rogue waves appearsometimes very often!!!!
• Origin deterministic (at least in our
  experiments)
• Different types of chaos without and with rogue
  waves
• Simple models allow heuristic interpretation for
  the generation of rogue waves

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Université de Nice Sophia Antipolis - CNRS
I N L N
I hope you enjoyed the presentation

• If not, please .do not kill me!!
• If Yes,
• Thank you

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Mode Locked TiSa Laser
. LB pump focusing lens R laser rod (L4mm)
M mirrors P1, 2 pair of fused silica prisms to
introduce negative GVD. The observations are done
with a fast photodiode (100 ps risetime) and a
350 MHz, 5 Gs/s digital oscilloscope with a
memory of 16 MB.
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Results
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Two chaotic regimes
P2
P1
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Statistics of pulse amplitude

• (a) Experimental, regime P2, 2?AI394 9978
  pulses, 237 are above the 2?AI value and 206 are
  above the 4? value. Note the L-shape. Optical
  rogue waves are hence observed.
• (b) Experimental, regime P1, 2?AI 417.6, 4?
  256 3747 pulses, the highest one has amplitude
  234? 2?AI and 4?.

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Model based on a five dim. map

• (c) Numerical, regime P2, 2?AI 56.8 4? 3?104
  pulses, 147 are above the 2?AI and 4?.
• (d) Numerical, regime P1, 2?AI 50.22, 4?
  48.25 104 pulses, the highest one is 27

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Theoretical results
(dE/dt) k ( 1 ia ) (N - 1)E I w E
Einj (dN/dt) g ( m N N E 2 )
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About Physical Origin (PRA to be published)
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Crisis
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