Waves Called Solitons: Concepts and Experiments PDF
Preview Waves Called Solitons: Concepts and Experiments
WavesCalledSolitons Springer-Verlag Berlin Heidelberg GmbH ONLINEL1BRARY Physics andAstronomy http://www.springer.de/phys/ Advanced Textsin Physics This program of advanced texts covers abroad spectrum oftopics which are of currentandemerginginterestinphysics.Eachbookprovidesacomprehensiveand yetaccessibleintroductiontoafieldatthe forefront ofmodernresearch.Assuch, these textsareintendedforseniorundergraduateand graduatestudentsattheMS and PhDlevel;however,researchscientists seekingan introductionto particular areas ofphysicswillalsobenefit from the titles inthis collection. Michel Remoissenet Waves Called Solitons Concepts and Experiments 3rd Revised and EnlargedEdition With 172Figures Springer ProfessorDr,Michel Remoissenet UniversiredeBourgogne Fac.desSeiencesandTechniques LaboratoiredePhysique 9,AvenueAlainSavary 21011DijonCedex,France E-mail:[email protected] LibraryofCongressCataloging-in-PublicationData. Remoissenet, M.,Wavescalledsolitons:conceptsand experiments1MichelRemoissenet.-jrdre.and enl.ed.p.cm.Includesbiliographicalreferencesand index. ISBN3-540-65919-6 1.Solitons. 1.Titel. QCI74.26.W28R46 1999 531.'1133-dc2l 99-23783 CIP ThirdEdition1999 CorrectedSecond Printing2003 ISBN978-3-642-08519-2 ISBN978-3-662-03790-4(eBook) DOI 10.1007/978-3-662-03790-4 Thiswork issubject to copyright.Allrights are reserved,whetherthe wholeor partofthe material isconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broad casting,reproduction on microfilmor in any otherway,and storagein data banks.Duplication of thispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLaw ofSeptember9,1965,in its currentversion, and permissionfor use must alwaysbe obtainedfrom Springer-Verlag.ViolationsareliableforprosecutionundertheGermanCopyrightLaw, Springer-VerlagBerlinHeidelbergNewYork amemberofBertelsmannSpringerScience+BusinessMediaGmbH http://www.springer.de ©Springer-VerlagBerlinHeidelberg1993,1996,1999 OriginallypublishedbySpringer-VerlagBerlinHeidelbergNewYorkin1999. Softcoverreprintofthehardcover3rdedition1999 Theuseofgeneraldescriptivenames, registered names, trademarks,etc.inthispublicationdoes not imply,evenintheabsence ofaspecificstatement,that such namesareexemptfrom therelevant pro tectivelawsandregulationsandthereforefreeforgeneral use. Typesettingbytheauthor,FinalprocessingbyperformHeidelbergSpringerTEXmacro package Coverdesign:design&productionGmbH,Heidelberg Printedonacid-freepaper 55/3141/mf 54 3210 Preface to the Corrected Printing The three years that have intervened since the appearance ofthe third edition of this book have witnessed some interesting developments in the field ofsoliton likeexcitations. Thepresentcorrected andrevisededition retains allthematerial ofthethird edition. There are five new short sections on compact-like modes, experiments and instabilities in optical systems, generation and detection ofdiscrete breathers in real lattices, acoustic solitons in lattices and excitable systems with nonlinear friction. Each of these sections refers to many relevant papers that can be consulted forfurther study. My particular thanks go to H.Jauslin who helped me to correct the manu script and to ex-student J. C. Comte whose research efforts are responsible for somepartsofthenewmaterial addedtothisedition. Again, for this edition I have benefited from the technical assistance of V.BoudonandP.Michaux. Dijon MichelRemoissenet November 2002 v Preface to the Third Edition In the third edition the presentation of new topics has been deliberately kept simpleforpedagogicalpurposes. Chapter 1wascompleted by references to thetidal boreand magnetic envelope solitons. Two new sections devoted to magnetic envelope solitons and signal processing with solitons have been added to Chap. 4. Shortcomments on models describing blood-pressure pulse propagation in terms of solitons were added to chapter 5. A description of a new mechanical transmission line with two equilibrium states has been included in Chap. 6. Such an analog device is useful to illustrate the properties of kink-solitons and to observe solitary waves with a compact shape,calledcompactons. Newreferences conceming recentadvances in experimental techniques and lattice effects were added to Chap.7. In Chap. 8 a short introduction to spatial optical solitons in continuous and discrete systems was included. Chapter 9 was completed by new sections devoted to nonlinear lattice models and energy localization. The concepts of self-trapped states and intrinsic localized modes or discrete breathers are dicussed. Different analog chains which make it possible to observe the characteristic features of discrete breathers,aredescribed. Contrary to previous chapters where we have considered reversible or conservative systems where solitons can exist owing to the dynamical balance between lineardispersion and nonlinearity, Chapter 11which isa new chapter, is devoted to irreversible systems where nonlinearity can balance the effects of dissipation leading to "diffusing solitary waves" or "diffusive solitons". We introduce this robust entity,which indeed is not a strict soliton, by considering a nonlinear electricaltransmission line,wherekink-shaped diffusivesolitons canbe predicted, and we present recent experiments. Then, we introduce reaction diffusion processes by considering a chemical model for first order non equilibrium phase transitions. In this model the powerbalancedsolitary wave or ditiusivesoliton represents amovingdomainwall orinterfacelayerbetween two phases. Wethenpresent anexperimentalelectricallatticewhichhas beenused as an analog model for the transmission of information along nerve fibers. In the next section we present a mechanical analog which makes it possible to observe diffusive solitons.The last sectionisdevotedto a discussionofreaction-diffusion processes inlatticesandpropagation failurephenomenon. Dijon MichelRemoissenet May 1999 VII Preface to the First Edition Nonlinearity is a fascinating element of nature whose importance has been appreciated for many years when considering large-amplitude wave motions observed in various fields ranging from fluids and plasmas to solid-state, chemical, biological, and geological systems. Localized large-amplitude waves called solitons, which propagate without spreading and have particle-like properties, represent one ofthe most striking aspects ofnonlinear phenomena. Although awealth ofliterature onthesubject, including theoretical and numerical studies, is available in good recent books and research journals, very little material has found its way into introductory texbooks and curricula. This is perhaps due to a beliefthat nonlinear physics is difficult and cannot be taught at an introductory level to undergraduate students and practitioners. Consequently, thereisconsiderable interest indevelopingpractical material suitable for students, atthelowestintroductory level. This book is intended to be an elementary introduction to the physics of solitons, for students, physicists, engineers and practitioners. We present the modeling ofnonlinearphenomenawhere soliton-like waves areinvolved,together with applications to a wide variety of concrete systems and experiments. This book is designed as abook ofphysical ideas and basic methods and not as an up to-the-minute book concerned with the latest research results.The background in physics and the amount of mathematical knowledge assumed of the reader is within thatusuallyaccumulatedbyjuniororsenior students inphysics. Much of the text of this book is an enlargement of a set of notes and descriptions of laboratory experiments developed over aperiod of years to supplement lectures onvarious aspects ofwave motion.Inspiteofthediversity of the material, the book is not a collection of disconnected topics, written for specialists. Instead, I have tried to supply the practical and fundamental background in soliton physics, and to plan the book in order that it should be as much aspossible aself-containedandreadableinterdisciplinarywhole.Often, the important ideas or results are repeated several times, in different contexts. Many ofmy choices ofemphasis and examples have been made with experimental aspectsin mind Several experiments describedin thisbookcanbeperfonnedby the resder. Although numerical studies play an important role in nonlinear science, I will not consider them in this book because they are described in a considerable bodyofliterature. IX In order to facilitate the use of this book, many illustrations have been included in the text and the details of theoretical calculations are relegated to appendicesat the end of each chapter.A number ofbasicreferencesare given as weIlasreferencesintendedtodocumentthehistoricaldevelopmentofthesubject. The referencing is not systematic; the bibliography listed at the end ofthe book serves onlyto advisethe readerwhichsources couldbe usedto fill ingaps in his orherbasicknowledgeandwhereheorshecouldturnforfurtherreading. Thetext is organizedas follows.Our introduction inChap.l is devotedto thebeautifulhistoricalpathofthesoliton.The fundamentalideasofwavemotion are then set forth in Chap.2 using simple electrical transmission lines and electrical networksas examples.At an elementary level,we reviewand illustrate the main properties oflinear nondispersive and dispersive waves propagating in one spatial dimension. In Chap.3, we consider waves in transmission lines with nonlinearity. These simple physical systems are very useful for a pedagogical introductionto the soliton concept, and they are easy to construct and to model, allowing one to become quickly familiar with the essential aspects of solitary waves and solitons, and their properties. Specifically, we firstexamine the effect ofnonlinearityon the shapeofa wavepropagating alonga nonlinear dispersion less transmission line. Then we consider the remarkable esse where dispersion and nonlinearity can balance to produce a pulse-Iike wave with a permanent protile. We describe simple experiments on pulse solitons, which illustrate the important features of such remarkable waves. In Chap.4 we consider the lattice solitons,whichcanpropagate onanelectricalnetwork;thenweexamine periodic wavetrains, and modulated waves such as envelope orhole solitons, which can travelalongelectricaltransmissionlines. Chapter 5 concentrates on such spectacular waves as the hydrodynamic pulse soliton, which was first observedin the nineteenth century,and the hydro dynamicenvelopesoliton.Simplewater-tankexperimentsaredescribed. In Chap. 6, by using a chain of coupled pendulums, that is, a mechanical transmission line, we introduce a new class of large amplitude waves, known as kink solitons and breather solitons which present remarkable particle-like properties.Simpleexperimentsthatallowoneto studyqualitativelytheproperties ofthesesolitonsarepresented. Chapter 7 deals with a more sophisticated device: the superconductive Josephson junction. Here the physical quantity of interest is a quantum of magnetic flux, or fluxon, which behaves like a kink soliton and has properties remarkably similar to the mechanical solitons of Chap.5. In Chap.8 brightand dark solitons emerge, which correspond to the optical envelope or optical hole solitons, respectively. They can be observed in optical fibers where exploitation of the typical dispersive and nonlinear effects has stimulated theoretical and experimentalstudiesonnonlinearguidedwaves. Whereas the previous chapters are concerned with solitons in the macro world, Chap.9 deals with nonlinear excitations in the microworld. Specifically, we consider the soliton concept in the study of nonlinear atomic lattices. The nonlinear equations that are encountered in the soliton story and models of several systems described in the text can be solved by using remarkable and x powerful mathematical techniques, the main steps ofwhich are given in the last chapter. If a substantial fraction of users of this book feel that it helped them to approach the fascinating world ofnonlinear waves or enlarge their outlook, its purpose willhavebeen fulfilled.Ihopethereader will feelencouraged tobring to mynotice anyremaining errors andothersuggestions. I have greatly benefited from frequent discussions with my colleagues and students. I am particularly grateful to Jean Marie Bilbault, who went over the entiremanuscript andgavemeinvaluablecomments. I would also like to thank Alwyn Scott whose criticism and suggestions helpedrefine themanuscript. I also wish to extend my appreciation to Patrick Marquie, Guy Millot, Jean Franccis Paquerot, Michel Peyrard, and Claudine and Gerard Pierre for their comments on various chapters. Special thanks go to Bemard Michaux for his assistance in designing and performing experiments, and improving numerous illustrations throughout the book. Finally, it is a pleasure for me to acknowledge the technical assistance I have received from Dominique Amoult and Claudine Jonon. Dijon MichelRemoissenet October 1993 XI Contents 1 Basic Conceptsand theDiscoveryofSolitons 1 1.1 Alookatlinear and nonlinearsignatures 1 1.2 Discovery ofthe solitarywave 3 1.3 Discovery of the soliton 7 1.4 The solitonconceptinphysics 11 2 LinearWavesin ElectricalTransmissionLines 12 2.1 Linear nondispersivewaves 12 2.2 Sinusoidal-wavecharacteristics 15 2.2.1 Waveenergydensity andpower 18 2.3 Thegroup-velocityconcept 19 2.4 Lineardispersivewaves 21 2.4.1 Dispersivetransmissionlines 21 2.4.2 Electricalnetwork 23 2.4.3 The weaklydispersive limit 26 2.5 Evolutionofawavepacket envelope 27 2.6 Dispersion-induced wavepacketbroadening 31 Appendix 2A. Generalsolutionfor theenvelope evolution 34 Appendix 2B. Evolutionoftheenvelope ofaGaussianwavepacket.. 35 3 Solitonsin NonlinearTransmissionLines 37 3.1 Nonlinear anddispersionless transmissionlines 37 3.2 Combined effectsofdispersionand nonlinearity 41 3.3 Electricalsolitary wavesand pulse solitons 42 3.4 Laboratoryexperimentson pulse solitons 46 3.4.1 Experimentalarrangement 46 3.4.2 Seriesofexperiments 48 3.5 Experimentswith a pocket versionof theelectricalnetwork 52 XIII
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