THE FRENKEL-KONTOROVA MODEL
INTRODUCTION THE FRENKEL-KONTOROVA MODEL THE SINE-GORDON EQUATION PHYSICAL MODELS GENERAL APPROACH A MECHANICAL MODEL DISLOCATION DYNAMICS SURFACES AND ADSORBED ATOMIC LAYERS INCOMMENSURATE PHASES IN DIELECTRICS CROWDIONS AND LATTICE DEFECTS MAGNETIC CHAINS JOSEPHSON JUNCTIONS NONLINEAR MODELS OF THE DNA DYNAMICS 2 10 HYDROGEN-BONDED CHAINS 2 11 MODELS OF INTERFACIAL SLIP KINKS THE PEIERLS-NABARRO POTENTIAL DYNAMICS OF KINKS E?ECTIVE EQUATION OF MOTION MOVING KINKS TRAPPED KINKS MULTIPLE KINKS GENERALIZED ON-SITE POTENTIAL BASIC PROPERTIES KINK INTERNAL MODES NONSINUSOIDAL ON-SITE POTENTIAL MULTIPLE-WELL POTENTIAL MULTI-BARRIER POTENTIAL DISORDERED SUBSTRATES POINT DEFECTS EXTERNAL INHOMOGENEOUS FORCE ANHARMONIC INTERATOMIC INTERACTION SHORT-RANGE INTERACTION NONCONVEX INTERATOMIC POTENTIALS KAC-BAKER INTERACTION LONG-RANGE INTERACTION COMPACTON KINKS BREATHERS PERTURBED SINE-GORDON BREATHERS LARGE-AMPLITUDE BREATHERS SMALL-AMPLITUDE BREATHERS BREATHER COLLISIONS MANY-SOLITON E?ECTS FRACTAL SCATTERING SOLITON COLD GAS IMPURITY MODES STRUCTURE AND STABILITY SOLITON INTERACTIONS WITH IMPURITIES DISCRETE BREATHERS GENERAL REMARKS EXISTENCE AND STABILITY THE DISCRETE NLS EQUATION DARK BREATHERS ROTOBREATHERS TWO-DIMENSIONAL BREATHERS PHYSICAL SYSTEMS AND APPLICATIONS GROUND STATE FIXED-DENSITY CHAIN COMMENSURATE CON?GURATIONS INCOMMENSURATE CON?GURATIONS FREE-END CHAIN FRANK-VAN-DER-MERWE TRANSITION DEVIL’S STAIRCASE AND PHASE DIAGRAM GENERALIZATIONS OF THE FK MODEL ON-SITE POTENTIAL OF A GENERAL FORM ANHARMONIC INTERATOMIC POTENTIAL NONCONVEX INTERACTION STATISTICAL MECHANICS INTRODUCTORY REMARKS GENERAL FORMALISM WEAK-BOND LIMIT: GLASS-LIKE PROPERTIES ISING-LIKE MODEL CON?GURATIONAL EXCITATIONS TWO-LEVEL SYSTEMS AND SPECI?C HEAT STRONG-BOND LIMIT: GAS OF QUASIPARTICLES SHARING OF THE PHASE SPACE AND BREATHERS KINK-PHONON INTERACTION KINK-KINK INTERACTION DISCRETENESS E?ECTS STATISTICAL MECHANICS OF THE FK CHAIN TRANSFER-INTEGRAL METHOD THE PSEUDO-SCHR¨ODINGER EQUATION SUSCEPTIBILITY HIERARCHY OF SUPERKINK LATTICES EQUAL-TIME CORRELATION FUNCTIONS GENERALIZED FK MODELS THERMALIZED DYNAMICS BASIC CONCEPTS AND FORMALISM BASIC FORMULAS MORI TECHNIQUE DI?USION COE?CIENTS NONINTERACTING ATOMS INTERACTING ATOMS DI?USION OF A SINGLE KINK LANGEVIN EQUATION INTRINSIC VISCOSITY ANOMALOUS DI?USION KINK DI?USION COE?CIENT DYNAMIC CORRELATION FUNCTIONS MASS TRANSPORT PROBLEM DI?USION IN A HOMOGENEOUS GAS APPROXIMATE METHODS PHENOMENOLOGICAL APPROACH SELF-DI?USION COE?CIENT PROPERTIES OF THE DI?USION COE?CIENTS DRIVEN DYNAMICS NONLINEAR RESPONSE OF NONINTERACTING ATOMS OVERDAMPED CASE UNDERDAMPED CASE OVERDAMPED FK MODEL DRIVEN KINK INSTABILITY OF FAST KINKS SUPERSONIC AND MULTIPLE KINKS LOCKED-TO-SLIDING TRANSITION COMMENSURATE GROUND STATES COMPLEX GROUND STATES AND MULTISTEP TRANSITION HYSTERESIS TRA?C JAMS 8 10 PERIODIC FORCES: DISSIPATIVE DYNAMICS 8 11 PERIODIC DRIVING OF UNDERDAMPED SYSTEMS RATCHETS PRELIMINARY REMARKS DI?ERENT TYPES OF RATCHETS SUPERSYMMETRY DI?USIONAL RATCHETS INERTIAL RATCHETS SOLITONIC RATCHETS SYMMETRY CONDITIONS ROCKED RATCHETS PULSATING RATCHETS EXPERIMENTAL REALIZATIONS FINITE-LENGTH CHAIN 10 1 GENERAL REMARKS 10 2 GROUND STATE AND EXCITATION SPECTRUM 10 2 1 STATIONARY STATES 10 2 2 CONTINUUM APPROXIMATION 10 2 3 DISCRETE CHAINS 10 2 4 VIBRATIONAL SPECTRUM 10 3 DYNAMICS OF A FINITE CHAIN 10 3 1 CATERPILLAR-LIKE MOTION 10 3 2 ADIABATIC TRAJECTORIES 10 3 3 DI?USION OF SHORT CHAINS 10 3 4 STIMULATED DI?USION 10 4 NONCONVEX POTENTIAL TWO-DIMENSIONAL MODELS 11 1 PRELIMINARY REMARKS 11 2 SCALAR MODELS 11 2 1 STATISTICAL MECHANICS 11 2 2 DYNAMIC PROPERTIES 11 3 ZIGZAG MODEL 11 3 1 GROUND STATE 11 3 2 AUBRY TRANSITIONS 11 3 3 CLASSI?CATION OF KINKS 11 3 4 ZIGZAG KINKS 11 3 5 APPLICATIONS 11 4 SPRING-AND-BALL VECTOR 2D MODELS 11 4 1 THE GROUND STATE 11 4 2 EXCITATION SPECTRUM 11 4 3 DYNAMICS 11 5 VECTOR 2D FK MODEL 11 5 1 LOCKED-TO-SLIDING TRANSITION 11 5 2 “FUSE-SAFETY DEVICE” ON AN ATOMIC SCALE CONCLUSION HISTORICAL REMARKS REFERENCES