40771-5 Braun U. 31.10.2003 9:05 Uhr Seite 1 TMP H K S 6 K iB O. M. Braun v H sr ha K O.M.Braun · Y.S.Kivshar u Y. S. Kivshar S a The Frenkel--Kontorova Model rn 4 1 This book presents an overview of the basic 5 1 6 concepts,methods and applications of nonlinear 0 The Frenkel-- 5 low-dimensional solid-state physics based on the 0 Frenkel--Kontorova model and its generalizations. 3 0 It allows a nonspecialist to acquire foundations Kontorova Model .1 in the interdisciplinary concepts and methods of CT 0 both solid-state physics and nonlinear science. oh .0 ne 3 It covers many important topics such as the non- ce F Concepts, Methods, de linear dynamics of discrete systems,the dynamics pre sig t n of solitons and their interaction,commensurate sn a ,k nd and incommensurate systems,statistical me- Me and Applications pro el d cehquaniliicbsr ioufmn odnylninaemairc ssy osfteimntse,rraacttcihnegt sm,aannyd- nboond-y tho--Ko uction systems.Also treated are the principal nonlinear dsn Gm equations along with the properties of their local- ,ato bH ized solutions and methods for their analysis. nr – B do e Ava nde p r pM licationodel das endgültigDieser Farbau s e Druckesdruck/p ISSN 0172-5998 rgebndf-file ISBN3-540-40771-5 is w ka ienn derge nur a bn 123 en nä !he 9 783540 407713 rn d › springeronline.com Texts and Monographs in Physics SeriesEditors: R.Balian,Gif-sur-Yvette,France W.Beiglbo¨ck,Heidelberg,Germany H.Grosse,Wien,Austria E.H.Lieb,Princeton,NJ,USA N.Reshetikhin,Berkeley,CA,USA H.Spohn,Mu¨nchen,Germany W.Thirring,Wien,Austria 3 Berlin Heidelberg NewYork HongKong London Milan Paris Tokyo O. M. Braun Y. S. Kivshar The Frenkel–Kontorova Model Concepts, Methods, and Applications With161Figures 1 3 ProfessorOlegM.Braun UkrainianAcademyofSciences InstituteofPhysics NaukiProspekt46 252650Kiev,Ukrainia ProfessorYuriS.Kivshar AustralianNationalUniversity OpticalSciencesCenter 0200CanberraACT,Australia LibraryofCongressCataloging-in-PublicationData Braun,O.M.(Oleg),1949– TheFrenkel-Kontorovamodel:concepts,methods,andapplications/O.M.Braun,Y.S. Kivshar. p.cm.–(Textsandmonographsinphysics,ISSN0172-5998) Includesbibligraphicalreferencesandindex. ISBN3-540-40771-5(alk.paper) 1.Solidstatephysics.2.Nonlineartheories. I.Kivshar,Y.S.(YuriS.),1959– II. Title,III.Series. QC176.B682003 530.4’1–dc22 2003059072 ISSN0172-5998 ISBN3-540-40771-5 Springer-VerlagBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialisconcerned, specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,reproductiononmicrofilm orinanyotherway,andstorageindatabanks.Duplicationofthispublicationorpartsthereofispermittedonlyunder theprovisionsoftheGermanCopyrightLawofSeptember9,1965,initscurrentversion,andpermissionforusemust alwaysbeobtainedfromSpringer-Verlag.ViolationsareliableforprosecutionundertheGermanCopyrightLaw. Springer-VerlagBerlinHeidelbergNewYork amemberofBertelsmannSpringerScience+BusinessMediaGmbH http://www.springer.de ©Springer-VerlagBerlinHeidelberg2004 PrintedinGermany Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply,eveninthe absenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprobreaktectivelawsandregulationsand thereforefreeforgeneraluse. Typesettingandproduction: Coverdesign:design&productionGmbH,Heidelberg Printedonacid-freepaper 55/3141/Ba 543210 For Our Parents and Families Preface Theoretical physics deals with physical models. The main requirements for a good physical model are simplicity and universality. Universal models which can be applied to describe a variety of different phenomena are very rare in physics and, therefore, they are of key importance. Such models attract the special attention of researchers as they can be used to describe underlying physical concepts in a simple way. Such models appear again and again over the years and in various forms, thus extending their applicability and educa- tional value. The simplest example of this kind is the model of a pendulum; this universal model serves as a paradigm which encompasses basic features of various physical systems, and appears in many problems of very different physical context. Solids are usually described by complex models with many degrees of freedom and, therefore, the corresponding microscopic equations are rather complicated. However, over the years a relatively simple model, known these daysastheFrenkel-Kontorovamodel,hasbecomeoneofthefundamentaland universal tools of low-dimensional nonlinear physics; this model describes a chain of classical particles coupled to their neighbors and subjected to a pe- riodicon-sitepotential.Althoughlinkswiththeclassicalformulationarenot oftenstatedexplicitlyindifferentapplications,manykindsofnonlinearmod- els describing the dynamics of discrete nonlinear lattices are based, directly orindirectly,ona1938classicalresultofFrenkelandKontorova,whoapplied a simple one-dimensional model for describing the structure and dynamics of a crystal lattice in the vicinity of a dislocation core. This is one of the firstexamplesinsolidstatephysicswhenthedynamicsofanextendeddefect in a bulk was modelled by a simple one-dimensional model. Over the years, similar ideas have been employed in many different physical problems, also providing a link with the mathematical theory of solitons developed later for the continuum analog of the Frenkel-Kontorova (FK) model. Inthecontinuumapproximation,theFKmodelisknowntoreducetothe exactly integrable sine-Gordon (SG) equation, and this explains why the FK model has attracted much attention in nonlinear physics. The SG equation gives an example of a fundamental nonlinear model for which we know ev- erythingaboutthedynamicsofnonlinearexcitations,namelyphonons,kinks (topological solitons), and breathers (dynamical solitons); and their multi- VIII Preface particle dynamics determines the global behavior of a nonlinear system as a whole. Although the FK model is inherently discrete and is not integrable, one may get a deep physical insight and simplify one’s understanding of the nonlinear dynamics using the language of the SG nonlinear modes as weakly interacting effective quasi-particles. The discreteness of the FK model mani- festsitselfinsuchphenomenaastheexistenceofaneffectiveperiodicenergy known as the Peierls-Nabarro potential. The simplicity of the FK model, due to the assumptions of linear inter- atomicforcesandasinusoidalexternalpotential,aswellasitssurprisingrich- ness and capability to describe a range of important nonlinear phenomena, has attracted a great deal of attention from physicists working in solid-state physicsandnonlinearscience.Manyimportantphysicalphenomena,ranging from solitons to chaos as well as from the commensurate-incommensurate phases to glass-like behavior, present complicated sub-fields of physics each requiringaspecialbook.However,theFKmodelprovidesauniqueopportu- nity to combine many such concepts and analyze them together in a unified and consistent way. The present book aims to describe, from a rather general point of view, the basic concepts and methods of low-dimensional nonlinear physics on the basis of the FK model and its generalizations. We are not restricted by the details of specific applications but, instead, try to present a panoramic view on the general properties and dynamics of solid state models and summarize the results that involve fundamental physical concepts. Chapter 1 makes an introduction into the classical FK model, while Chap. 2 discusses in more detail the applicability of the FK model to dif- ferent types of physical systems. In Chap. 3 we introduce one of the most important concepts, the concept of kinks, and describe the characteristics of the kink motion in discrete chains, where kinks are affected by the Peierls- Nabarro periodic potential. In Chap. 4 we analyze another type of nonlinear mode, the spatially localized oscillating states often called intrinsic localized modes or breathers. We show that these nonlinear modes may be understood asageneralizationoftheSGbreathersbutexistinthecaseofstrongdiscrete- ness. Chapters 3 and 4 also provide an overview of the dynamical properties of the generalized FK chains which take into account more general types of on-site potential as well as anharmonic interactions between particles in the chain. The effect of impurities on the dynamics of kinks as well as the dy- namics and structure of nonlinear impurity modes are also discussed there. Chapter 5 gives a simple introduction to the physics of commensurate and incommensurate systems, and it discusses the structure of the ground state of the discrete FK chain. We show that the FK model provides probably the simplest approach for describing systems with two or more competing spa- tialperiods.Whiletheinteractionbetweentheatomsfavorstheirequidistant separation with a period corresponding to the minimum of the interatomic potential, the interaction of atoms with the substrate potential (having its Preface IX own period) tends to force the atoms into a configuration where they are regularlyspaced.InChap.5weemploytwomethodsfordescribingtheprop- ertiesoftheFKmodel:first,inthecontinuumapproximationwedescribethe discrete model by the exactly integrable SG equation, and second, we study theequationsforstationaryconfigurationsofthediscreteFKmodelreducing ittotheso-calledstandard map,oneoftheclassicalmodelsofstochasticthe- ory.ThestatisticalmechanicsoftheFKmodelisdiscussedinChap.6,which also includes the basic results of the transfer-integral method. Here, the FK model again appears to be unique because, on the one hand, it allows the derivationofexactresultsintheone-dimensionalcaseand,ontheotherhand, itallowsfortheintroductionofweaklyinteractingquasi-particles(kinksand phonons) for describing the statistical mechanics of systems of strongly in- teracting particles. Chapter 7 gives an overview of the dynamical properties of the FK model at nonzero temperatures, including kink diffusion and mass transport in nonlinear discrete systems. Chapter 8 discusses the dynamics of nonlinear chains under the action of dc and ac forces when the system is far from its equilibrium state. Chapter 9 discuses ratchet dynamics in driven systems with broken spatial or temporal symmetry when a directed motion is induced. The properties of finite-length chains are discussed in Chap. 10, whereastwo-dimensionalgeneralizationsoftheFKmodelareintroducedand described in Chap. 11, for both scalar and vector models. In the concluding Chap. 12 we present more examples where the basic concepts and physical effects, demonstrated above for simple versions of the FK chain, may find applicationsinabroadercontext.Atlast,thefinalchapterincludessomein- terestinghistoricalremarkswrittenbyProf.AlfredSeeger,oneofthepioneers in the study of the FK model and its applications. Wethankourmanycolleaguesandfriendsaroundtheglobewhohavecol- laborated with us on different problems related to this book, or contributed toourunderstandingofthefield.Itisimpossibletolistallofthem,butweare particularlyindebtedtoA.R.Bishop,L.A.Bolshov,D.K.Campbell,T.Daux- ois, S.V. Dmitriev, S. Flach, R.B. Griffiths, B.A. Ivanov, A.M. Kosevich, A.S.Kovalev,I.F.Lyuksyutov,B.A.Malomed,A.G.Naumovets,M.V.Paliy, M. Peyrard, M. Remoissenet, J. Ro¨der, A. Seeger, S. Takeno, L.-H. Tang, A.V. Ustinov, I.I. Zelenskaya, and A.V. Zolotaryuk. Canberra, Australia Oleg Braun May 2003 Yuri Kivshar