Table Of ContentLecture Notes in Physics 926
Miguel Onorato
Stefania Residori
Fabio Baronio Editors
Rogue and
Shock Waves
in Nonlinear
Dispersive
Media
Lecture Notes in Physics
Volume 926
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Miguel Onorato (cid:129) Stefania Residori (cid:129) Fabio Baronio
Editors
Rogue and Shock Waves
in Nonlinear Dispersive
Media
123
Editors
MiguelOnorato StefaniaResidori
DipartimentodiFisica InstitutNonLinéairedeNice
UniversitàdiTorino CentreNationaldelaRecherche
Torino,Italy Scientifique,CNRS
UniversitédeNice-SophiaAntipolis
Valbonne,France
FabioBaronio
DipartimentodiIngegneriadell’Informazione
UniversitàdiBrescia
Brescia,Italy
ISSN0075-8450 ISSN1616-6361 (electronic)
LectureNotesinPhysics
ISBN978-3-319-39212-7 ISBN978-3-319-39214-1 (eBook)
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Preface
Rogueanddispersiveshockwavesarephenomenathatoccurinnonlineardispersive
media. They have been studied in different fields of physics, including ocean
waves,nonlinearoptics,Bose-Einsteincondensates,plasma physics,etc. Theyare
apparentlyvery differentphenomena;however,they are both characterisedby the
developmentofextremes:whiletheamplitudeofaroguewavereacheslargevalues,
shockwavesdevelopextremegradients.Thepresenceofrogueandshockwavesin
anincoherentwavesystemdrasticallyinfluencesitsstatisticalproperties.
Duringthelast15years,thefieldof“roguewaves”hasexperiencedaveryquick
development. The original motivation of the research was related mostly to the
occasional measurementof extreme waves on the surface of the ocean and to the
occurrenceofvariousaccidentscausedbytheimpactoflargeamplitudewaveson
ships. Only morerecently,it hasbeen shownthat extremelightfluctuationscould
be observedin an opticalfibre; the research activity has now broadenedup to the
creationofanewfieldinitsown.
A rogue wave is the manifestation of a processof focussing of energy.One of
the most accredited explanations of the formation of rogue waves (at least in an
idealisedcase)isthemodulationalinstabilityprocessbywhichasmallperturbation
of a plane wave can grow exponentially fast in time. This mechanism has been
knownfromthelatesixties,butonlyatthebeginningofthenewcentury,ithasbeen
associatedtoroguewaves.Thenonlinearstagesofthemodulationalinstabilityare
describedbyexactsolutionsofthenonlinearSchrödingerequation.Thosesolutions,
named breathers, have been considered as the prototypes of rogue waves in the
ocean.
Thedevelopmentofaninfinitegradientinfinitetime(gradientcatastrophe)with
consequentwave-breaking is probably a less baleful phenomenon,nonetheless of
extreme nature.Mathematically,a classical shock wave is a discontinuous(weak)
solutiondescribingpropagationbeyondabreakingpointwhereaninfinitederivative
develops.Startingfromthelastcentury,inseveralbranchesofphysics,ithasbeen
recognised that the dissipation plays an important role in regularising the jump
leading to physical shock waves characterised by strong but finite gradients. A
completely different non-trivial dynamics may result when dispersion dominates.
v
vi Preface
In the latter case, after wave-breaking, the infinite gradient is regularised by the
spontaneous onset of fast non-stationary oscillations that progressively fill an
extended region. The effect of dispersion on the regularisation of shock waves is
veryintriguing,leadingtoacomplexstronglynon-stationarydynamics.Opticshas
providedonlyrecentlytheopportunitytoobservethesedynamicsthatappeartobe
similartothoseproducedinhydrodynamicsunderspecificconditionsinvolving,for
examples,strongtidalborespropagatingupstreamin riverestuaries. Shockwaves
haveimpactonmanypracticalsituationsrangingfromphotonicstohydraulicdam-
breakingtotrafficorgasdynamicsproblems,forwhichthedispersiveeffects,which
arenormallyneglected,maydetermineaqualitativechangeofthesystembehaviour.
In the summer of 2015, we have organised a school on rogue and dispersive
shockwavesinthebeautifulvillageofCargese,Corsica(France).Theideawasto
bringtogethertop-leveltheoreticalphysicists,mathematiciansandexperimentalists
workingmainlyinoceanwavesandnonlinearopticswiththeaimofpresentingto
students and young researchers a unifying concept of rogue and dispersive shock
waves. The school lasted for 2 weeks: the first one was characterised by a set
of 3-h lectures whose goal was to introduce the students to the deterministic and
statistical approach to the subject in the various fields. During the second week,
shorter talks, in the workshop format, were given in which more advanced topics
were discussed. It turned out that the event was very successful with about 70
(including students and lecturers) participants and many useful discussions. The
present book can be considered as a collection of notes from some of the 3-h
lectures.Itincludesafirstchapter“HydrodynamicandOpticalWaves:ACommon
ApproachforUnidimensionalPropagation”inwhichacloseanalogybetweenoptics
andhydrodynamicwavesismade.Thechapterintroducesthereadertothenonlinear
Schrödinger(NLS)equationwhichhasplayedamajorroleintheunderstandingof
rogue waves. A second chapter “Integrability in Action: Solitons, Instability and
RogueWaves” is devoted to the role played by integrable equations in the devel-
opment of the field; the chapter explains how to construct solutions that describe
coherentstructuressuchassolitonsandroguewavesorhowtoinvestigatepatterns
asthosecausedbyshockwavesorinstabilities.The thirdchapter“Hydrodynamic
Envelope Solitons and Breathers” and fourth chapter “Experiments on Breathers
in Nonlinear Fibre Optics” describe experiments in hydrodynamicsand nonlinear
optics where exact breather solutions of the NLS equation have been reproduced
experimentally.Thefollowingthreechaptersaredevotedtoastatisticaldescription
of rogue waves in water waves: in chapter “Hamiltonian Description of Ocean
Waves and Freak Waves” a theory for estimating the kurtosis and the skewness
of the surface elevation from wave spectra is explained. The theory has a major
relevance in the forecasting of rogue waves in operational systems. Its validation
withfieldmeasurementsisalsoreported.Inchapter“ModellingTransientSeaStates
with the Generalised Kinetic Equation” an extension of the theory presented in
chapter“HamiltonianDescriptionofOceanWavesandFreakWaves”isdiscussed:
in particular, the role of sharp changes of wind in the generation of rogue waves
is highlighted. Chapter “Rogue Waves in Random Sea States: An Experimental
Perspective”describestheresultsfromanumberofexperimentsperformedinwave
Preface vii
tanks with the aim of establishing the probability of formation of rogue waves
in different sea states; experiments including currents under the waves are also
described.Inchapters“IntroductiontoWaveTurbulenceFormalismsforIncoherent
OpticalWaves”and“IntegrableTurbulencewithNonlinearRandomOpticalWaves”
theattentionisturnedtothedescriptionofincoherentopticalwaves.Thereaderis
brought to the construction of statistical tools for describing a system of a large
numberofinteractingopticalwaves;issuesrelatedtocondensation,thermalisation,
incoherent modulational instability and wave turbulence are discussed. Chapter
“Integrable Turbulence with Nonlinear Random Optical Waves” is related to the
emerging field of integrable turbulence, i.e. the nonlinear state generated by a
largenumberof incoherentwavesdescribedby integrableequations.Experiments
in optical fibres ruled by NLS equation and numerical simulations revealing the
formationof heavy tails in the probabilitydensity functionof the wave amplitude
aredescribed.Thelasttwochaptersdealwithdispersiveshockwaves:thefirstofthe
two includesa pedagogicalintroductionto the Whitham modulationequationthat
playsamajorroleintheunderstandingofdispersiveshockwaves.Thelastchapter
includes experimental results in optics and hydrodynamics displaying dispersive
shockwaves.
Our idea was to create a book accessible to graduate students and researchers
workingin variousfields ofphysicsandappliedmathematics.Moreover,we hope
thatthisworkmightbeusefultostudentsbybringingtotheirattentionproblemsof
fundamentalnature that are often neglected in graduatecourses. The bookcannot
be considered as exhaustive; the reason is that the field on rogue and dispersive
shockwavesisrapidlyevolving,andeverymonth,newinterestingideasappearin
theliterature.We havemadea selectionofthe topics,givingprioritiesto whatwe
believearetheresultsdescribedbyacommonandinterdisciplinarylanguage.Each
chapterisself-consistentanditdoesnotrequirethereadingofthepreviousone.
We would like to thank all the authors of the chapters of the book and, more
generally,allthespeakersoftheschoolinCargesewithwhomwehaveexchanged
many fruitful and interesting discussions on rogue and dispersive shock waves.
Finally, we would also like to acknowledge the CNRS, the Università di Torino,
the Università di Brescia and the European Geophysical Union for their financial
supporttotheschool.
Torino,Italy MiguelOnorato
Brescia,Italy FabioBaronio
Valbonne,France StefaniaResidori
February2016
Contents
HydrodynamicandOpticalWaves:A CommonApproach
forUnidimensionalPropagation ............................................... 1
Miguel Onorato, Fabio Baronio, Matteo Conforti,
AminChabchoub,PierreSuret,andStephaneRandoux
1 Introduction.................................................................... 2
2 NormalVariablesintheWaveEquation ..................................... 2
3 Water Waves in One Horizontal Dimension and Their
HamiltonianFormulation..................................................... 5
3.1 SurfaceGravityWaves:TheCanonicalTransformation.............. 7
3.2 TheNLSEquationforSurfaceGravityWavesinInfinite
WaterDepth............................................................. 9
4 OpticalWavesinNormalVariables .......................................... 11
4.1 Three-WaveInteractions:(cid:2).2/ Media .................................. 13
4.2 Four-WaveInteractionsinPure(cid:2).3/Media............................ 16
4.3 Four-WaveMixingina(cid:2).2/ and(cid:2).3/ Medium......................... 18
4.4 TheStokesExpansioninOpticalWaves............................... 19
5 DiscussionandConclusions.................................................. 20
References......................................................................... 21
IntegrabilityinAction:Solitons,InstabilityandRogueWaves............. 23
AntonioDegasperisandSaraLombardo
1 IntroductiontoIntegrabilityandSolitons.................................... 23
2 IntegrabilityinAction:TheNLSEquationasStudyCase.................. 28
2.1 ConservationLawsfromtheLaxPair.................................. 28
2.2 TheInitialValueProblemandParticularSolutions................... 31
3 NLSEquation:LinearInstabilityandRogueWaves ........................ 36
4 WaveCoupling:IntegrabilityandRogueWaves............................. 40
5 IntegrabilityinAction:BeyondtheNLSModel............................. 46
References......................................................................... 48
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