Nonlinear Waves and Solitons on Contours and Closed Surfaces PDF
Preview Nonlinear Waves and Solitons on Contours and Closed Surfaces
Springer Series in Synergetics SeriesEditors HenryD.I.Abarbanel,InstituteforNonlinearScience,UniversityofCalifornia, SanDiego,CA,USA DanBraha,NewEnglandComplexSystemsInstitute,Cambridge,MA,USA PéterÉrdi,CenterforComplexSystemsStudies,KalamazooCollege,USA HungarianAcademyofSciences,Budapest,Hungary KarlJFriston,InstituteofCognitiveNeuroscience,UniversityCollegeLondon, London,UK HermannHaken,CenterofSynergetics,UniversityofStuttgart,Stuttgart,Germany ViktorJirsa,CentreNationaldelaRechercheScientifique(CNRS),Universitédela Méditerranée,Marseille,France JanuszKacprzyk,SystemsResearch,PolishAcademyofSciences,Warsaw,Poland KunihikoKaneko,ResearchCenterforComplexSystemsBiology,TheUniversity ofTokyo,Tokyo,Japan ScottKelso,CenterforComplexSystemsandBrainSciences,FloridaAtlantic University,BocaRaton,FL,USA JürgenKurths,NonlinearDynamicsGroup,UniversityofPotsdam,Potsdam, Germany RonaldoMenezes,ComputerScienceDepartment,UniversityofExeter,Exeter, UK AndrzejNowak,DepartmentofPsychology,WarsawUniversity,Warsaw,Poland HassanQudrat-Ullah,DecisionSciences,YorkUniversity,Toronto,ON,Canada LindaReichl,CenterforComplexQuantumSystems,UniversityofTexas,Austin, TX,USA FrankSchweitzer,SystemDesign,ETHZurich,Zurich,Switzerland DidierSornette,EntrepreneurialRisk,ETHZurich,Zurich,Switzerland StefanThurner,SectionforScienceofComplexSystems,MedicalUniversityof Vienna,Vienna,Austria Editor-in-Chief PeterSchuster,TheoreticalChemistryandStructuralBiology,Universityof Vienna,Vienna,Austria SpringerSeriesinSynergetics FoundingEditor:H.Haken TheSpringerSeriesinSynergeticswasfoundedbyHermanHakenin1977.Since then,theserieshasevolvedintoasubstantialreferencelibraryforthequantitative, theoreticalandmethodologicalfoundationsofthescienceofcomplexsystems. Throughmanyenduringclassictexts,suchasHaken’sSynergeticsandInforma- tion and Self-Organization, Gardiner’s Handbook of Stochastic Methods, Risken’s The Fokker Planck-Equation or Haake’s Quantum Signatures of Chaos, the series hasmade,andcontinuestomake,importantcontributionstoshapingthefoundations ofthefield. The series publishes monographs and graduate-level textbooks of broad and generalinterest,withapronouncedemphasisonthephysico-mathematicalapproach. Andrei Ludu Nonlinear Waves and Solitons on Contours and Closed Surfaces Third Edition AndreiLudu DepartmentofMathematics Embry-RiddleAeronauticalUniversity DaytonaBeach,FL,USA ISSN 0172-7389 ISSN 2198-333X (electronic) SpringerSeriesinSynergetics ISBN 978-3-031-14640-4 ISBN 978-3-031-14641-1 (eBook) https://doi.org/10.1007/978-3-031-14641-1 1stand2ndeditions:©Springer-VerlagBerlinHeidelberg2007,2012 3rdedition:©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringer NatureSwitzerlandAG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Tomyfamily,themostimportant presence. Foreword ThestoryofsolitarywavestracesbacktoJohnScottRussel.Approaching200years agohewrote: Iwasobservingthemotionofaboatwhichwasrapidlydrawnalonganarrowchannelby apairofhorses,whentheboatsuddenlystopped—notsothemassofwaterinthechannel whichithadputinmotion;itaccumulatedroundtheprowofthevesselinastateofviolent agitation,thensuddenlyleavingitbehind,rolledforwardwithgreatvelocity,assumingthe formofalargesolitaryelevation,arounded,smoothandwell-definedheapofwater,which continueditscoursealongthechannelapparentlywithoutchangeofformordiminutionof speed.Ifolloweditonhorseback,andovertookitstillrollingonatarateofsomeeight orninemilesanhour,preservingitsoriginalfiguresomethirtyfeetlongandafoottoa footandahalfinheight.Itsheightgraduallydiminished,andafterachaseofoneortwo milesIlostitinthewindingsofthechannel.Such,inthemonthofAugust1834,wasmy firstchanceinterviewwiththatsingularandbeautifulphenomenonwhichIhavecalledthe WaveofTranslation. Russelwentontoconductexperimentsandpublishedhisfindingsin1845(check this).Initially,majorfiguressuchasStokesandAirydeniedtheexistenceofwhatwe wouldnowcallatravelingwaveonthesurfaceofwaterinachannel.Inthesecond halfofthenineteenthcentury,oneseesinthecorrespondencebetweenStokesand Raleigh that Stokes had changed his mind and this fact even appears in published work.Intheperiodofthiscorrespondence,Rayleighfoundanapproximaterelation betweentheamplitudeandspeedofasolitarywaveinachannel.However,itwas lefttoBoussinesqinthe1870stowritedownevolutionequationsthatapproximated themotionofdisturbancesonthesurfaceofwaterandwhichfeaturedexactsolitary- wavesolutions.OneofthesewasthecelebratedKorteweg-deVriesequationofwater wavetheorythatwasrederivedbyJosephKortewegandhisstudentGustavdeVries in1895.Theissueofexistenceoftheseso-calledsolitarywaveshavingbeensettled, atleastasfarasthenineteenthcenturyhydrodynamicistswereconcerned,thesubject wentmoribund. Itcamebacktolife,thoughindisguise,inworkofFermi,Pasta,UlamandTsingou onalatticeandspringmodelforheatconductioninthe1950s.Later,bytakingan appropriate continuum limit of this mass and spring model, Kruskal and Zabusky cameagaintotheKorteweg-deVriesequation.Thistime,however,thesubjectdid vii viii Foreword not die. In 1967, the inverse scattering theory for this equation was discovered by Gardner, Greene, Miura and Kruskal. Peter Lax took the first step in putting this formalism into a very imaginative mathematical structure. Since then, the subject rapidlyachievedindustrialproportions,withtensofthousandsofjournalpagesand withmany,manyapplicationsofthetheory. AsAndreiLudu,theauthorofthepresentmonographwritesinhisintroduction, consideringthelargeliteratureonsolitarywaves,whyyetanotherbook?Thereare severalthingsthatsetthistextapartfromothersinthefield.Firstistheoverallfocus upon solitary waves defined on compact spaces. Of course, one thinks initially of theclassicalcnoidal-wavesolutionsoftheKorteweg-deVriesequation,butasLudu ably shows, this is the tip of a very large iceberg. Another aspect of the text that strikesanewchordisthedifferentialgeometricperspective; theviewthatsolitary waves can be realized as the motion of a planar or three-dimensional curve under particularflowconditionsandwithsuitableinitialconditions.Thisisnotoriginalto the text in question, but an overall assessment of these ideas and a comprehensive reviewofitsapplicationsisnottobefoundelsewhereintheliterature.And,speaking of applications, the text ends with a large number of very diverse and interesting applications. Thetextbreaksintofourparts.PartsIandII,whichcomprisethefirsteightchap- ters,containasketchoftherelevanttopologyandespeciallythedifferentialgeometry of curves and surfaces in two and three spatial dimensions. It should be acknowl- edgedthatthismaterialisnotforbeginners.Someonewithoutpriorknowledgeofat leastportionsofthismaterialwillnotfinditeasygoing.However,asareminderto thosewithsomeknowledge,andafocusonexactlywhatisneededfromdifferential geometryinwhatfollows,itisveryhelpful.EspeciallythematerialinChapter6will beusefulevenforthecognoscenti. Chapter7worksouttheconnectionbetweenthemotionofcurvesintwoandthree dimensions and integrable systems. Chapter 8 does the same thing for the motion ofsurfaces.Technically,thisistheheartofthescript.Thiswillbenewmaterialto manyreaders;indeed,itisadevelopingsubjectinthemathematicalfirmament. Ludu’sexpositioninPartsIandIIistechnicallysound,butitmakesmuchofits headwaybywayofappealingtoourintuition.Noteverytheoremisprovedindetail, whichisquiteokaygiventheoverallgoalofthetext. InPartsIIIandIV,thetextbecomesmoreconcrete.Itbeginswithamoreorless standarddiscussionofthekinematicsoffluidmotioninChapter9.Knowledgeable readersmaywellskipthis,butforfolksalittlerusty,itishelpful.Someofthenotation islaidoutinthischapteraswell. Chapters10and11findusderivingtheEulerandNavier-Stokesequations.This includesaverydetaileddiscussionofsurfacetensionfromageometricalperspective. Hegoesontoderivemanyofourfavoriteapproximatemodels,suchastheKorteweg- deVriesequation,themodifiedKorteweg-deVriesequation,theBoussinesqequation and the cubic Schrodinger equation. He examines the well-known solitary-wave solutionsoftheseequationsbywayofthemathematicalstructuredevelopedinPart I.HealsoderiveswhathetermstheGKdVequation(GeneralizedKorteweg-deVries equation)thatresultsfromcarryingouttheformalasymptoticsintheshallowwater Foreword ix parameterandthenonlinearparametertohigherorder.Thisequationspecializesto the various more familiar equations. Again, what is distinctly non-standard is his concentrationuponsolitarywavesdefinedoncompactspacesthatcanbeobtained viathemotionofcurveswhosetheorywasdevelopedinPartII.Thispartisalsonot forabeginner.Withoutpriorbackgroundinthesesortsofderivations,itwillbehard going.Hardgoing,butworththeeffort. Chapters12–15mightwellhavebeenlumpedintoPartIIofthetext.Whilethey enlarge upon the theory, they emerge from physical considerations. Chapters 12– 14areconcernedwiththefascinatingshapeoscillationsofliquiddropsintwoand three space dimensions. Chapter 15 presents another quite different point of view thatyieldssomeofthesamefascinatingshapesthatappearedearlierindroplets. Inthefourthportionofthetext,Ludushowshisscientificupbringing.Hestarted life as a physicist and throughout his career he has been closely tied to real-world phenomena.HeadmirablyshowsoffhisbreadthinPartIVofthetext.Herewefind himdealingwithawholestableofsolitonsthatariseinsomeunlikelyplaces.There are solitons on filaments of various sorts, solitons on stiff chains, solitons on the boundariesofmicroscopicstructures,solitonsatstellarscales. Thetextfinisheswithamathematicalannexthatincludessomeinterestingremarks thatdidn’tfitanywhereelseinthetext. Thisbookisnottobereadinanarmchair.AsLudustatesinhisopeningremarks, itismeanttobestudiedwithpencilandpaperathandandwithanalgebraicmanip- ulation program up on the screen of a computer. It is a text dense with ideas and methods, both mathematical and scientific, and a serious addition to the literature. Thefactthatitisgoingintoathirdeditionatteststoitsimpact. Chicago,USA HongqiuChen JerryBona Preface to the Third Edition Inordertoofferasmuchcontentaspossiblefromallchaptersofthebooktoreaders with various prerequisites in mathematics, we present below a reader’s map that canhelpreaderstonavigatethroughthebookwithoutbeingstuckinsectionswith densermathematicalcontent.Prettymuchlikeonaskiingcourse,weintroducethree possiblepathstomeettheinterestofallourreaders: • No*asteriskisthepaththatdoesn’trequestspecialprerequisitesinmathematics, exceptcalculusandfirstlevelcourseinmathematicalphysics.Forthesereaders, werecommendthefollowingpath: Introduction →2.1→3.1→3.2→3.3→3.12→3.13→4.1→4.2→ 5→6.1→6.5→7.1→7.3→9.1→9.3→9.5→9.6.1→10.1→10.2→ 10.3→10.4.1→10.5→10.6.1→11.1→12.1→12.6→ 13.1→14.1→14.2→14.3→17.3→18.2→18.3→19 • Sections labeled with one asterisk ∗ request some previous knowledge in real analysis, differential systems and elements of geometry. For these readers we recommendinadditiontothe"No∗asteriskpath"toaddthefollowingsections: 3.4→3.5→3.6→3.7→3.10→6.3→6.4→7.5→9.4→9.6.2→ 10.4.3→10.6.2→11.2→11.5→12.2→12.3→12.4→12.5→12.6→ 13→14.4→14.5→15.2→18.1→18.4. xi
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