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.

Crisis at the Origin of Crisis at the Origin of Deterministic Rogue WavesDeterministic Rogue WavesPPME, Universite de la Nouvelle CaledoniePPME, Universite de la Nouvelle Caledonie

C. Metayer, A. Serres, J. Tredicce INLN, UMR 6618 UNS-CNRS FranceINLN, UMR 6618 UNS-CNRS France

S. Barland, M. GiudiciCEILAP - CITEDEF – ArgentinaCEILAP - CITEDEF – Argentina

A. Hnilo, M. KovalskiA. Hnilo, M. KovalskiUniv. Politecn. Cataluna, SpainUniv. Politecn. Cataluna, Spain

Masoller, C.Univ. Fed Pernambuco, Recife, PE BrazilUniv. Fed Pernambuco, Recife, PE Brazil

W. Barbosa, F. Menezes D’Aguiar,J. Rios Leite, Rosero E.

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

Is it true? Are rogue waves so Is it true? Are rogue waves so common?common?

This fact is in contradiction with the Gaussian models This fact is in contradiction with the Gaussian models used to describe fluctuations of the wave height in the used to describe fluctuations of the wave height in the sea*. sea*.

* M. S. Longuet-Higgins, Phil. Trans. Roy. Soc. A * M. S. Longuet-Higgins, Phil. Trans. Roy. Soc. A 249249 321 (1957).321 (1957).

S. Aberg and G. Lindgren, Height distribution of S. Aberg and G. Lindgren, Height distribution of stochastic Lagrange ocean waves, Prob. Eng. Mech. stochastic Lagrange ocean waves, Prob. Eng. Mech. 2323, 359 (2008), 359 (2008)

HOWEVER……HOWEVER……

Ferry rescue after freak wave in Irish Sea

The freighter Riverdance was hit by a giant wave during severe gales in the Irish Sea…..

But….What is the definition of a But….What is the definition of a rogue wave?rogue wave?

Old Recipe: Take the 1/3 biggest amplitude Old Recipe: Take the 1/3 biggest amplitude waves; calculate their average value; waves; calculate their average value; multiply by 2….whatever amplitude exceeds multiply by 2….whatever amplitude exceeds such value is a rogue wave!!!such value is a rogue wave!!!

More Recent Recipe: Take the probability More Recent Recipe: Take the probability distribution; calculate distribution; calculate ; multiply by 4; ; multiply by 4; whatever….; whatever….;

and if you want a BIG BIG rogue and if you want a BIG BIG rogue wave…multiply by 8 wave…multiply by 8

In the WEB:

It 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.

Some Bibliography about Rogue Some Bibliography about Rogue WavesWaves

Osborne, A.R. et al.; Phys. Lett. A 275, 386 Osborne, A.R. et al.; Phys. Lett. A 275, 386 (2000); and PRL 96, 014503 (2006).(2000); and PRL 96, 014503 (2006).

Clauss, G.F.; Appl. Ocean Res. 24, 147 (2002) 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).

How was that “Opticians” got How was that “Opticians” got interested on Rogue Waves?interested on Rogue Waves?

A “NONLINEAR OPTICS PHYSICIST” WENT TO A “NONLINEAR OPTICS PHYSICIST” WENT TO THE IRISH PUB….and then some papers appear THE IRISH PUB….and then some papers appear in Nature or other “GO..O..D” Journalsin Nature or other “GO..O..D” Journals

D. R. Solli, C. Ropers et al, Optical rogue waves, D. R. Solli, C. Ropers et al, Optical rogue waves, Nature 450 1054 (2007).Nature 450 1054 (2007).

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev and J. M. Dudley, The Genty, N. Akhmediev and J. M. Dudley, The Peregrine soliton in nonlinear fibre optics Nature Peregrine soliton in nonlinear fibre optics Nature Phys. 6, 790 (2010).Phys. 6, 790 (2010).

A. Montina, U. Bortolozzo, S. Residori, F.T. Arecchi, A. Montina, U. Bortolozzo, S. Residori, F.T. Arecchi, Phys. Rev. Lett. 103, 173901 (2009)Phys. Rev. Lett. 103, 173901 (2009)

Our First Experiments….Our First Experiments….

1) Mode Locked Ti:Sa laser1) Mode Locked Ti:Sa laser

Hnilo et al. (Opt. Lett. November 2011)Hnilo et al. (Opt. Lett. November 2011)

2) Semiconductor Laser with Injected Signal2) Semiconductor Laser with Injected Signal

Bonatto et al. (PRL, July 2011)Bonatto et al. (PRL, July 2011)

3) Laser with saturable absorber (Journal of 3) Laser with saturable absorber (Journal of Optics, submitted)Optics, submitted)

Laser with Injected SignalLaser with Injected Signal

Probability distribution of maxima Probability distribution of maxima

Our Already Published Our Already Published ConclusionsConclusions

1) Extreme Events are rare but they can be 1) Extreme Events are rare but they can be much more probable than in Gaussian much more probable than in Gaussian models when the dynamical behavior is models when the dynamical behavior is “Deterministically” Chaotic“Deterministically” Chaotic

2) There is “chaos” without rogue waves 2) There is “chaos” without rogue waves and chaos with rogue wavesand chaos with rogue waves

Some questions:Some questions:

How? How? What is the dynamical process the What is the dynamical process the laser use to generate “extreme events”?laser use to generate “extreme events”?

Can we predict deterministic extreme events Can we predict deterministic extreme events in optical systems?in optical systems?

Can we control them? Can we control them?

How?How?

a) Intermittency ….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??

Bifurcation Diagrams

Experimental results Experimental results

Laser with Modulated parameterLaser with Modulated parameterRemembering very old « times »:Remembering very old « times »:H.G. Solari J, E. Eschenazi, R. Gilmore H.G. Solari J, E. Eschenazi, R. Gilmore et al., Opt. et al., Opt. Commun. 64, 49 (1987)Commun. 64, 49 (1987)

onon

““Crisis of chaotic attractors”Crisis of chaotic attractors”

Two ingredients: 1) chaosTwo ingredients: 1) chaos

2) Enough low dissipation in order to have 2) Enough low dissipation in order to have “generalized” multistability (several stable “generalized” multistability (several stable dynamical solutions for the same parameter values) dynamical solutions for the same parameter values)

Crisis of chaotic attractorsCrisis of chaotic attractors

External crisis in a laser with External crisis in a laser with mopdulated parametermopdulated parameter

Then extreme events appear Then extreme events appear after an external crisisafter an external crisis

Predicting “Rogue waves”? 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

ConclusionsConclusions

1)1) External crisis produce abrupt expansion of External crisis produce abrupt expansion of chaotic attractors and are at the origin of chaotic attractors and are at the origin of some extreme eventssome extreme events

2)2) Deterministic extreme events could be Deterministic extreme events could be predicted with « some » anticipationpredicted with « some » anticipation

3)3) I still do not know if we are able to control I still do not know if we are able to control deterministic extreme events deterministic extreme events

BUTBUT

I am always looking for the rogue I am always looking for the rogue waves in New Caledoniawaves in New Caledonia

Laser with saturable absorber in Laser with saturable absorber in Q-switch regime (to be subm. to Q-switch regime (to be subm. to

special issue)special issue) With Alejandro Hnilo and Marcelo Kovalski, With Alejandro Hnilo and Marcelo Kovalski, CEILAP, Villa Martelli, ArgentinaCEILAP, Villa Martelli, Argentina

Relevance of Spatial EffectsRelevance of Spatial Effects

Theoretical results without spatial Theoretical results without spatial effectseffects

Number of rogue waves in parameter space in LIS (from J. Number of rogue waves in parameter space in LIS (from J. Zamora)Zamora)

Some bibliography to take into Some bibliography to take into account:account:

V. Balakrishnan, C. Nicolis, and G. Nicolis; V. Balakrishnan, C. Nicolis, and G. Nicolis; ««Extreme Value Distributions in Chaotic Extreme Value Distributions in Chaotic Dynamics” Dynamics” J. of Stat.Phys. 80, 307 1995J. of Stat.Phys. 80, 307 1995

C. Nicolis,V. Balakrishnan, and G. Nicolis C. Nicolis,V. Balakrishnan, and G. Nicolis “Extreme Events in Deterministic Dynamical “Extreme Events in Deterministic Dynamical Systems” Systems” PRL PRL 97, 97, 210602 (2006)210602 (2006)

P. Gaspard and X.J. Wang “Sporadicity: between P. Gaspard and X.J. Wang “Sporadicity: between periodic and chaotic dynamical behaviors” Proc. periodic and chaotic dynamical behaviors” Proc. Nat. Acad. Sci. USA 85, 4591 (1988).Nat. Acad. Sci. USA 85, 4591 (1988).

PerspectivesPerspectives

1) Experiment of laser with modulation in solid 1) Experiment of laser with modulation in solid state laser (at CEILAP). Why solid state and not state laser (at CEILAP). Why solid state and not semiconductor at INLN?semiconductor at INLN?

2) Experiments laser with injection large Fresnel 2) Experiments laser with injection large Fresnel number (if INLN agree)number (if INLN agree)

3) large fresnel number edge emitter lasers 3) large fresnel number edge emitter lasers (UFPE)(UFPE)

4) laser with feedback (UPC) + theory4) laser with feedback (UPC) + theory 5) Numerical work at UNC5) Numerical work at UNC

ConclusionsConclusions

Rogue waves appear……sometimes very Rogue waves appear……sometimes very often!!!!often!!!!

Origin: deterministic (at least in our Origin: deterministic (at least in our experiments)experiments)

Different types of chaos: without and with Different types of chaos: without and with rogue wavesrogue waves

Simple models allow heuristic interpretation Simple models allow heuristic interpretation for the generation of rogue wavesfor the generation of rogue waves

Université de Nice Sophia Antipolis - CNRS I N L N

I hope you enjoyed the I hope you enjoyed the presentationpresentation

If not, please If not, please ….do not kill ….do not kill me!!me!!

If Yes, If Yes,

Thank you Thank you

Mode Locked Ti:Sa LaserMode Locked Ti:Sa Laser

. LB: pump focusing lens; R: laser rod (L=4mm); 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.

Results:Results:

Two chaotic regimes:Two chaotic regimes:

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(a) Experimental, regime P2, 2(a) Experimental, regime P2, 2AI=394; 9978 pulses, 237 are above AI=394; 9978 pulses, 237 are above the 2the 2AI value and 206 are above the 4AI value and 206 are above the 4 value. Note the L-shape. value. Note the L-shape. Optical rogue waves are hence observed. Optical rogue waves are hence observed.

(b) Experimental, regime P1, 2(b) Experimental, regime P1, 2AI = 417.6, 4AI = 417.6, 4 = 256; 3747 pulses, = 256; 3747 pulses, the highest one has amplitude 234the highest one has amplitude 234 2 2AI and 4AI and 4..

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(c) Numerical, regime P2, 2(c) Numerical, regime P2, 2AI = 56.8 = 4AI = 56.8 = 4; ; 33104 pulses, 147 are above the 2104 pulses, 147 are above the 2AI and 4AI and 4. .

(d) Numerical, regime P1, 2(d) Numerical, regime P1, 2AI = 50.22, 4AI = 50.22, 4 = = 48.25; 104 pulses, the highest one is 2748.25; 104 pulses, the highest one is 27

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Theoretical resultsTheoretical results(dE/dt) = ( 1 + i ) (N - 1)E + I E + Einj

(dN/dt) = ( – N – N | E |2 )

About Physical Origin (PRA to be published)

CrisisCrisis