Nonlinear Waves and Solitons on Contours and Closed Surfaces PDF
Preview Nonlinear Waves and Solitons on Contours and Closed Surfaces
Springer Complexity Springer Complexity is an interdisciplinary program publishing the best research and academic-level teaching on both fundamental and applied aspects of complex systems — cutting across all traditional disciplines of the natural and life sciences, engineering, economics,medicine,neuroscience,socialandcomputerscience. Complex Systems are systems that comprise many interacting parts with the ability to generate anew qualityof macroscopic collectivebehavior themanifestations of whichare the spontaneous formation of distinctive temporal, spatial or functional structures. Models of such systems can be successfully mapped onto quite diverse “real-life” situations like theclimate,thecoherentemissionoflightfromlasers,chemicalreaction-diffusionsystems, biologicalcellularnetworks, thedynamicsofstockmarketsandoftheinternet,earthquake statistics and prediction, freeway traffic, the human brain, or the formation of opinions in socialsystems,tonamejustsomeofthepopularapplications. Although their scope and methodologies overlap somewhat, one can distinguish the following main concepts and tools: self-organization, nonlinear dynamics, synergetics, turbulence,dynamicalsystems,catastrophes,instabilities,stochasticprocesses,chaos,graphs and networks, cellular automata, adaptive systems, genetic algorithms and computational intelligence. 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EditorialandProgrammeAdvisoryBoard HenryAbarbanel,InstituteforNonlinearScience,UniversityofCalifornia,SanDiego,USA DanBraha,NewEnglandComplexSystemsInstituteandUniversityofMassachusettsDartmouth,USA Pe´ter E´rdi, Center for Complex Systems Studies, Kalamazoo College, USA and Hungarian Academy ofSciences,Budapest,Hungary KarlFriston,InstituteofCognitiveNeuroscience,UniversityCollegeLondon,London,UK HermannHaken,CenterofSynergetics,UniversityofStuttgart,Stuttgart,Germany ViktorJirsa,CentreNationaldelaRechercheScientifique(CNRS),Universite´ delaMe´diterrane´e,Marseille, France JanuszKacprzyk,SystemResearch,PolishAcademyofSciences,Warsaw,Poland KunihikoKaneko,ResearchCenterforComplexSystemsBiology,TheUniversityofTokyo,Tokyo,Japan ScottKelso,CenterforComplexSystemsandBrainSciences,FloridaAtlanticUniversity,BocaRaton,USA Markus Kirkilionis, Mathematics Institute and Centre for Complex Systems, University of Warwick, Coventry,UK Ju¨rgenKurths,NonlinearDynamicsGroup,UniversityofPotsdam,Potsdam,Germany LindaReichl,CenterforComplexQuantumSystems,UniversityofTexas,Austin,USA PeterSchuster,TheoreticalChemistryandStructuralBiology,UniversityofVienna,Vienna,Austria FrankSchweitzer,SystemDesign,ETHZurich,Zurich,Switzerland DidierSornette,EntrepreneurialRisk,ETHZurich,Zurich,Switzerland StefanThurner,SectionforScienceofComplexSystems,MedicalUniversityofVienna,Vienna,Austria Springer Series in Synergetics FoundingEditor:H.Haken TheSpringerSeriesinSynergeticswasfoundedbyHermanHakenin1977.Since then,theserieshasevolvedintoasubstantialreferencelibraryforthequantitative, theoreticalandmethodologicalfoundationsofthescienceofcomplexsystems. Throughmanyenduringclassictexts,suchasHaken’sSynergeticsandInforma- tion andSelf-Organization,Gardiner’sHandbookof StochasticMethods,Risken’s The FokkerPlanck-Equationor Haake’sQuantum Signaturesof Chaos, the series hasmade,andcontinuestomake,importantcontributionstoshapingthefoundations ofthefield. Theseriespublishesmonographsandgraduate-leveltextbooksofbroadandgen- eralinterest,withapronouncedemphasisonthephysico-mathematicalapproach. Forfurthervolumes: http://www.springer.com/series/712 Andrei Ludu Nonlinear Waves and Solitons on Contours and Closed Surfaces Second Edition 123 Dr.AndreiLudu Embry-RiddleAeronauticalUniversity DepartmentofMathematics S.ClydeMorrisBlvd.600 32114DaytonaBeachFlorida USA [email protected] ISSN0172-7389 ISBN978-3-642-22894-0 e-ISBN978-3-642-22895-7 DOI10.1007/978-3-642-22895-7 SpringerHeidelbergDordrechtLondonNewYork LibraryofCongressControlNumber:2011944331 (cid:2)c Springer-VerlagBerlinHeidelberg2012 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations areliabletoprosecutionundertheGermanCopyrightLaw. Theuseofgeneral descriptive names,registered names, trademarks, etc. inthis publication does not imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotective lawsandregulationsandthereforefreeforgeneraluse. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To Delia,Missy,andNana, foreverything. • Preface to the Second Edition EverythingthePoweroftheWorld doesisdoneinacircle.Theskyis roundandIhaveheardthattheearth isroundlikeaballandsoareallthestars. Thewind,initsgreatestpower,whirls. Birdsmaketheirnestsincircles, fortheirsisthesamereligionasours. Thesuncomesforthandgoesdown againinacircle.Themoondoesthe sameandbothareround.Eventhe seasonsformagreatcircleintheir changingandalwayscomebackagain towheretheywere.Thelifeofaman isacirclefromchildhoodtochildhood. Andsoitiseverythingwherepowermoves. BlackElk(1863–1950) Nonlinearphenomenarepresentintriguingandcaptivatingmanifestationsofnature. Thenonlinearbehaviorisresponsiblefortheexistenceofcomplexsystems,catas- trophes, vortex structures, cyclic reactions, bifurcations, spontaneous phenomena, phasetransitions,localizedpatternsandsignals,andmanyothers.Theimportance of studying nonlinearities has increased over the decades, and has found more andmorefields of applicationrangingfromelementaryparticles,nuclearphysics, biology, wave dynamics at any scale, fluids, plasmas to astrophysics. The soliton is the central character of this 167-year-old story. A soliton is a localized pulse traveling without spreading and having particle-like properties plus an infinite numberof conservationlaws associated to its dynamics. In general, solitons arise as exact solutions of approximative models. There are different explanation, at differentlevels,fortheexistenceofsolitons.Fromtheexperimentalistpointofview, solitons can be created if the propagation configuration is long enough, narrow enough (like long and shallow channels, fiber optics, electric lines, etc.), and the vii viii PrefacetotheSecondEdition surroundingmediumhasanappropriatenonlinearresponseprovidingacertaintype of balance between nonlinearity and dispersion. From the numerical calculations pointofview,solitonsarelocalizedstructureswithveryhighstability,evenagainst collisions between themselves. From the theory of differential equations point of view, solitons are cross-sections in the jet bundle associated to a bi-Hamiltonian evolution equation (here Hamiltonian pairs are requested in connection to the existence of an infinite collection of conservation laws in involution). From the geometry point of view, soliton equations are compatibility conditions for the existence of a Lie group. From the physicist point of view, solitons are solutions of an exactlysolvable modelhavingisospectral propertiescarryingoutan infinite numberofnonobviousandcounterintuitiveconstantsofmotion. The progress in the theory of solitons and integrable systems has allowed the study of many nonlinear problems in mathematics and physics: nonlocal interac- tions, collective excitations in heavy nuclei, Bose–Einstein condensates in atomic physics,propagationofnervouspulses,swimmingofmotilecells,nonlinearoscilla- tionsofliquiddrops,bubbles,andshells,vorticesinplasmaandinatmosphere,tides inneutronstars,onlytoenumeratefewofpossibleapplications.Anumberofother applicationsofsolitontheoryalsoleadtothestudyofthedynamicsofboundaries. Inthat,thelastthreedecadeshaveseenthecompletionofthefoundationforwhat today we call nonlinear contour dynamics. The subsequent stage of development alongthistopicwasconnectedwiththeconsiderationofanalmostincompressible systems,wheretheboundary(contourorsurface)playsthemajorrole. Many of the integrable nonlinear systems have equivalent representations in terms of differential geometry of curves and surfaces in space. Such geometric realizationsprovidenew insight into the structure of integrableequations, as well as new physical interpretations. That is why the theory of motions of curves and surfaces, including here filaments and vortices, represents an important emerging fieldformathematics,engineeringandphysics. The first problem about such compact systems is that shape solitons, which usually exist in infinite long and shallow propagationmedia, cannot survive on a circleorsphere.Thatisbecausesuchcompactmanifoldscannotoffertherequested typeofenvironment(longandnarrow),evenbytheintroductionofshallowlayers and rigid cores. However, there is another basic idea which supports, in a natural way, the existence of nonlinear solutions on compact spaces. Because of its high localization,asolitonisnotauniquesolutionforthepartialdifferentialsystem.Its positionin spaceis undeterminedbecause,farawayfromits center,theexcitation is practically zero. On the other hand, all linear equations provide uniqueness propertiesfortheirsolutions.Itresultsthatstronglylocalizedsolutions,andalmost compact supported solutions can be generated only within nonlinear equations. There is an exception here: the finite difference equations with their compact supportedwaveletsolutions,butinsomesenseafinitedifferenceequationissimilar toanonlineardifferentialone. Despite the many applications and publications on nonlinear equations on compactdomains,therearestillnobooksintroducingthistheory,exceptforseveral setsoflecturenotes.Onereasonforthismaybethatthefieldisstillundergoinga PrefacetotheSecondEdition ix major developmentand has not yet reached the perfection of a systematic theory. Another reason is that a fairly deep knowledge of integrable systems on compact manifoldshasbeenrequiredfortheunderstandingofsolitonsonclosedcurvesand compactsurfaces. The goal of the second edition of this book is to analyze the existence and describe the behavior of solitons traveling on closed, compact surfaces or curves. Theapproachofthephysicalproblemsrangingfromnucleartoastrophysicalscales is made in the language of differential geometry. The text is rather intended to be an introduction to the physics of solitons on compact systems like filaments, loops, drops, etc., for students, mathematicians, physicists, and engineers. The author assumes that the reader has some previous knowledge about solitons and nonlinearity in general. The book provide the reader examples of systems and models where the interaction between nonlinearities and the compact boundaries isessentialfortheexistenceandthedynamicsofsolitons. We focused on interesting and recent aspects of relations between integrable systemsandtheirsolutionsanddifferentialgeometry,mainlyoncompactmanifolds. Thebookconsistsof 17 chapters,a mathematicalannex,anda bibliography.First part contains the fundamental differential geometry and analysis approach. To renderthisbookaccessible to studentsin science andengineering,Chap.2 recalls somebasic elementsoftopologywith emphasisonthe conceptof beingcompact. In Chap.3 we review the representation formulas for different dimensions. The formulas express how a lot of information about the evolution of differentiable formsandfieldsinsideacompactdomaincanberecoveredonlyfromitsboundary. Chapter4 introducesthe readerto the calculuson differentiablemanifolds,vector fields,forms,andvarioustypesofderivatives.Wetakethereaderfrommapallthe way to the Poincare´ lemma. Next we introduce different types of fiber bundles, including the Cartan theory of frames, and the theory of connection and mixed covariant derivative (for immersions). Without always presenting the proofs, we triedthoughtokeepahighlevelofrigorousness(relyingonclassicalmathematical textbooks)all across the text while we still introduceintuitive commentsfor each definition or affirmation. Chapter 5 lays the basis for the differential geometry of curvesinR .Wedevoteherespecialsectionstoclosedcurvesandcurveslyingon 3 surfaces.Complementary,inChap.6weintroduceelementsofdifferentialgeometry ofthesurfaceswithapplicationstotheactionofdifferentialoperatorsonsurfaces. In Chap.7 we derivethe theoryof motionof curves, bothin two-dimensions,and inthegeneralcase.Wedevotedasectionontheaxiomaticdeductionofthetheory of motionsbased on differentiableformsand Cartan connectiontheory.We relate thesemotionswithsolitonsolutionsandfindthenonlinearintegrablesystemsthat canberepresentedbysuchmotionsofcurves.InChap.8wediscussthetheoryof motionofsurfaces,andwealsorelateittointegrablesystems. Thesecondpartofthemonographcontainsanexpositionofthebasicbranchesof nonlinearhydrodynamics.Theworkingframeofhydrodynamicsisthemaincontent ofthefirstpartofthemonograph,namelyChap.9.InChap.10wediscussproblems on surface tension effectsand representationtheoremsforfluid dynamicsmodels. Chapter 11 concentrates with one-dimensional integrable systems on compact
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