Rogue and Shock Waves in Nonlinear Dispersive Media PDF
Preview Rogue and Shock Waves in Nonlinear Dispersive Media
Lecture 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 FoundingEditors W.Beiglböck J.Ehlers K.Hepp H.Weidenmöller EditorialBoard M.Bartelmann,Heidelberg,Germany B.-G.Englert,Singapore,Singapore P.HaRnggi,Augsburg,Germany M.Hjorth-Jensen,Oslo,Norway R.A.L.Jones,Sheffield,UK M.Lewenstein,Barcelona,Spain H.vonLoRhneysen,Karlsruhe,Germany J.-M.Raimond,Paris,France A.Rubio,Hamburg,Germany M.Salmhofer,Heidelberg,Germany S.Theisen,Potsdam,Germany D.Vollhardt,Augsburg,Germany J.D.Wells,AnnArbor,USA G.P.Zank,Huntsville,USA The Lecture Notes in Physics The series Lecture Notes in Physics (LNP), founded in 1969, reports new devel- opmentsin physicsresearch and teaching-quicklyand informally,but with a high qualityand the explicitaim to summarizeand communicatecurrentknowledgein anaccessibleway.Bookspublishedinthisseriesareconceivedasbridgingmaterial between advanced graduate textbooks and the forefront of research and to serve threepurposes: (cid:129) to be a compact and modern up-to-date source of reference on a well-defined topic (cid:129) to serve as an accessible introduction to the field to postgraduate students and nonspecialistresearchersfromrelatedareas (cid:129) to be a source of advanced teaching material for specialized seminars, courses andschools Bothmonographsandmulti-authorvolumeswillbeconsideredforpublication. Editedvolumesshould,however,consistofaverylimitednumberofcontributions only.ProceedingswillnotbeconsideredforLNP. VolumespublishedinLNParedisseminatedbothinprintandinelectronicfor- mats,theelectronicarchivebeingavailableatspringerlink.com.Theseriescontent isindexed,abstractedandreferencedbymanyabstractingandinformationservices, bibliographicnetworks,subscriptionagencies,librarynetworks,andconsortia. Proposalsshouldbe sent to a memberof the EditorialBoard, ordirectly to the managingeditoratSpringer: ChristianCaron SpringerHeidelberg PhysicsEditorialDepartmentI Tiergartenstrasse17 69121Heidelberg/Germany [email protected] Moreinformationaboutthisseriesathttp://www.springer.com/series/5304 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) DOI10.1007/978-3-319-39214-1 LibraryofCongressControlNumber:2016950208 ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland 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 ix
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