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 D’Aguiar,
J. Rios Leite, Rosero E.
 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.
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……
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
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
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
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).
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)
Our First Experiments….
1) Mode Locked Ti:Sa 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)
Laser with Injected Signal
Probability distribution of maxima
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
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?
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??
Bifurcation Diagrams
Experimental results
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)
Crisis of chaotic attractors
External crisis in a laser with
mopdulated parameter
Then extreme events appear
after an external crisis
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
Conclusions
1) External crisis produce abrupt expansion of
chaotic attractors and are at the origin of
some extreme events
2) Deterministic extreme events could be
predicted with « some » anticipation
3) I still do not know if we are able to control
deterministic extreme events
BUT
I am always looking for the rogue
waves in New Caledonia
Laser with saturable absorber in
Q-switch regime (to be subm. to
special issue)
 With Alejandro Hnilo and Marcelo Kovalski,
 CEILAP, Villa Martelli, Argentina
Relevance of Spatial Effects
Theoretical results without spatial
effects
Number of rogue waves in parameter space in LIS (from J.
Zamora)
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).
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
Conclusions
 Rogue waves appear……sometimes 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
Université de Nice Sophia Antipolis - CNRS
I hope you enjoyed the
presentation
 If not, please
….do not kill
me!!
 If Yes,
Thank you
INLN
Mode 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:
1.0
1.0
0.8
0.8
Normalized Intensity
Normalized Intensity
Two chaotic regimes:
0.6
0.4
0.2
0.0
760
0.6
0.4
0.2
0.0
780
800
820
840
860
760
770
780
790
Wavelenght (nm)
Wavelenght (nm)
P1
P2
800
810
820
Statistics of pulse amplitude
500
a
2400
Number of pulses
Number of pulses
3200
1600
800
0
b
400
300
200
100
80
0
100
200
300
400
120
160
200
240
Detector signal amplitude
Detector signal amplitude
 (a) Experimental, regime P2, 2AI=394; 9978 pulses, 237 are above
the 2AI value and 206 are above the 4s value. Note the L-shape.
Optical rogue waves are hence observed.
 (b) Experimental, regime P1, 2AI = 417.6, 4s = 256; 3747 pulses,
the highest one has amplitude 234 2AI and 4s.
Model based on a five dim. map
1000
c
Number of pulses
Number of pulses
10000
1000
100
10
20
30
40
50
Detector signal amplitude
60
d
800
600
400
200
0
20
22
24
26
Detector signal amplitude
 (c) Numerical, regime P2, 2AI = 56.8 = 4s; 3104
pulses, 147 are above the 2AI and 4s.
 (d) Numerical, regime P1, 2AI = 50.22, 4s =
48.25; 104 pulses, the highest one is 27
Theoretical results
(dE/dt) = k ( 1 + ia ) (N - 1)E + I w E + Einj
(dN/dt) = g ( m – N – N | E |2 )
About Physical Origin (PRA to be
published)
Crisis