DISCRETE INTEGRABLE SYSTEMS

By X. GAVIS

INTRODUCTION THE PAINLEV´E PROPERTY AND THE NAIVE PAINLEV´E TEST FROM THE NAIVE TO THE POLY-PAINLEV´E TEST THE PAINLEV´E PROPERTY FOR THE PAINLEV´E EQUATIONS SATO THEORY AND TRANSFORMATION GROUPS A UNI?ED APPROACH TO INTEGRABLE SYSTEMS THE UNIVERSAL GRASSMANN MANIFOLD THE KP EQUATION PL¨UCKER RELATIONS THE KP EQUATION AS A DYNAMICAL SYSTEM ON A GRASSMANNIAN GENERALIZATION TO THE KP HIERARCHY WAVE FUNCTIONS, ?-FUNCTIONS AND THE BILINEAR IDENTITY PSEUDO-DI?ERENTIAL OPERATORS THE SATO EQUATION AND THE BILINEAR IDENTITY ?-FUNCTIONS AND THE BILINEAR IDENTITY TRANSFORMATION GROUPS THE BOSON-FERMION CORRESPONDENCE TRANSFORMATION GROUPS AND ?-FUNCTIONS B¨ACKLUND TRANSFORMATIONS FOR THE KP HIERARCHY EXTENSIONS AND REDUCTIONS EXTENSIONS OF THE KP HIERARCHY REDUCTIONS OF THE KP HIERARCHY SPECIAL SOLUTIONS DETERMINANT AND PFA?AN DE?NITION LINEARITY AND ALTERNATIVITY COFACTOR AND EXPANSION FORMULA ALGEBRAIC IDENTITIES GOLDEN THEOREM DI?ERENTIAL FORMULA DI?ERENCE FORMULAS DISCRETE WRONSKI PFA?ANS DISCRETE GRAM PFA?ANS DISCRETE BILINEAR EQUATIONS DISCRETE WRONSKI PFA?AN DISCRETE GRAM PFA?AN CONCLUDING REMARKS DISCRETE DI?ERENTIAL GEOMETRY INTEGRABILITY AS CONSISTENCY ORIGIN AND MOTIVATION: DI?ERENTIAL GEOMETRY EQUATIONS ON QUAD-GRAPHS INTEGRABILITY AS CONSISTENCY DISCRETE FLAT CONNECTIONS ON GRAPHS QUAD-GRAPHS 3D-CONSISTENCY ZERO-CURVATURE REPRESENTATION FROM THE 3D-CONSISTENCY CLASSI?CATION GENERALIZATIONS: MULTIDIMENSIONAL AND NON-COMMUTATIVE (QUANTUM) CASES YANG-BAXTER MAPS FOUR-DIMENSIONAL CONSISTENCY OF THREE-DIMENSIONAL SYSTEMS NONCOMMUTATIVE (QUANTUM) CASES SMOOTH THEORY FROM THE DISCRETE ONE DISCRETE LAGRANGIAN MODELS POISSON BRACKETS AND HAMILTONIAN FLOWS SYMPLECTIC MANIFOLDS POISSON REDUCTION COMPLETE INTEGRABILITY LAX REPRESENTATIONS LAGRANGIAN LAGRANGIAN MECHANICS ON LIE GROUPS INVARIANT LAGRANGIANS AND THE LIE–POISSON BRACKET 10 1 CONTINUOUS–TIME CASE 10 2 DISCRETE–TIME CASE LAGRANGIAN REDUCTION AND EULER–POINCAR´E EQUATIONS ON SEMIDIRECT PRODUCTS 11 1 CONTINUOUS–TIME CASE 11 2 DISCRETE–TIME CASE NEUMANN SYSTEM 12 1 CONTINUOUS–TIME DYNAMICS 12 2 B¨ACKLUND TRANSFORMATION FOR THE NEUMANN SYSTEM 12 3 RAGNISCO’S DISCRETIZATION OF THE NEUMANN SYSTEM 12 4 ADLER’S DISCRETIZATION OF THE NEUMANN SYSTEM GARNIER SYSTEM 13 1 CONTINUOUS–TIME DYNAMICS 13 2 B¨ACKLUND TRANSFORMATION FOR THE GARNIER SYSTEM 13 3 EXPLICIT DISCRETIZATION OF THE GARNIER SYSTEM MULTI–DIMENSIONAL EULER TOP 14 1 CONTINUOUS–TIME DYNAMICS 14 2 DISCRETE–TIME EULER TOP RIGID BODY IN A QUADRATIC POTENTIAL 15 1 CONTINUOUS–TIME DYNAMICS 15 2 DISCRETE–TIME TOP IN A QUADRATIC POTENTIAL MULTI–DIMENSIONAL LAGRANGE TOP 16 1 BODY FRAME FORMULATION 16 2 REST FRAME FORMULATION 16 3 DISCRETE–TIME ANALOGUE OF THE LAGRANGE TOP: REST FRAME FORMULATION 16 4 DISCRETE–TIME ANALOGUE OF THE LAGRANGE TOP: MOVING FRAME FORMULATION RIGID BODY MOTION IN AN IDEAL FLUID: THE CLEBSCH CASE 17 1 CONTINUOUS–TIME DYNAMICS 17 2 DISCRETIZATION OF THE CLEBSCH PROBLEM, CASE A = B2 17 3 DISCRETIZATION OF THE CLEBSCH PROBLEM, CASE A = B SYSTEMS OF THE TODA TYPE 18 1 TODA TYPE SYSTEM 18 2 RELATIVISTIC TODA TYPE SYSTEM BIBLIOGRAPHICAL REMARKS SYMMETRIES OF DISCRETE SYSTEMS COMMEN ORDINARY DI?ERENCE SCHEMES AND THEIR POINT SYMMETRIES ORDINARY DI?ERENCE SCHEMES POINT SYMMETRIES OF ORDINARY DI?ERENCE SCHEMES EXAMPLES OF SYMMETRY ALGEBRAS OF O?S LIE POINT SYMMETRIES OF PARTIAL DI?ERENCE SCHEMES PARTIAL DI?ERENCE SCHEMES SYMMETRIES OF PARTIAL DI?ERENCE SCHEMES THE DISCRETE HEAT EQUATION LORENTZ INVARIANT DI?ERENCE SCHEMES SYMMETRIES OF DISCRETE DYNAMICAL SYSTEMS GENERAL FORMALISM ONE-DIMENSIONAL SYMMETRY ALGEBRAS ABELIAN LIE ALGEBRAS OF DIMENSION N ?2 SOME RESULTS ON THE STRUCTURE OF LIE ALGEBRAS NILPOTENT NON-ABELIAN SYMMETRY ALGEBRAS SOLVABLE SYMMETRY ALGEBRAS WITH NON-ABELIAN NILRADICALS SOLVABLE SYMMETRY ALGEBRAS WITH ABELIAN NILRADICALS NONSOLVABLE SYMMETRY ALGEBRAS FINAL COMMENTS ON THE CLASSI?CATION GENERALIZED POINT SYMMETRIES OF LINEAR AND LINEARIZABLE SYSTEMS UMBRAL CALCULUS UMBRAL CALCULUS AND LINEAR DI?ERENCE EQUATIONS SYMMETRIES OF LINEAR UMBRAL EQUATIONS THE DISCRETE BURGERS EQUATION AND ITS SYMMETRIES DISCRETE PAINLEV´E EQUATIONS: A REVIEW THE (INCOMPLETE) HISTORY OF DISCRETE PAINLEV´E EQUATIONS DETECTORS, PREDICTORS, AND PROGNOSTICATORS (OF INTEGRABILITY) DISCRETE P’S GALORE INTRODUCING SOME ORDER INTO THE D-P CHAOS WHAT MAKES DISCRETE PAINLEV´E EQUATIONS SPECIAL? PUTTING SOME REAL ORDER TO THE D-P CHAOS MORE NICE RESULTS ON D-P’S EPILOGUE SPECIAL SOLUTIONS FOR DISCRETE PAINLEV´E EQUATIONS WHAT IS A DISCRETE THE CON AND THEIR SPECIAL SOLUTIONS SPECIAL FUNCTION SOLUTIONS FOR SYMMETRIC DISCRETE PAINLEV´E EQUATIONS THE CASE OF ASYMMETRIC DISCRETE PAINLEV´E EQUATIONS SOLUTIONS BY DIRECT LINEARISATION CONTINUOUS PAINLEV´E EQUATIONS SYMMETRIC DISCRETE PAINLEV´E EQUATIONS ASYMMETRIC DISCRETE PAINLEV´E EQUATIONS OTHER TYPES OF SOLUTIONS FOR D-P’S FROM ELEMENTARY TO HIGHER-ORDER SOLUTIONS AUTO-B¨ACKLUND AND SCHLESINGER TRANSFORMATIONS THE BILINEAR FORMALISM FOR D-PS THE CASORATI DETERMINANT SOLUTIONS BONUS TRACK: SPECIAL SOLUTIONS OF ULTRA-DISCRETE PAINLEV´E EQUATIONS ULTRADISCRETE SYSTEMS (CELLULAR AUTOMATA) BOX-BALL SYSTEM ULTRADISCRETIZATION BBS AS AN ULTRADISCRETE LIMIT OF THE DISCRETE KP EQUATION BBS AS ULTRADISCRETE LIMIT OF THE DISCRETE TODA EQUATION GENERALIZATION OF BBS BBS SCATTERING RULE AND YANG-BAXTER RELATION EXTENSIONS OF BBSS AND NON-AUTONOMOUS DISCRETE KP EQUATION FROM INTEGRABLE LATTICE MODEL TO BBS TWO-DIMENSIONAL INTEGRABLE LATTICE MODELS AND R-MATRICES CRYSTALLIZATION AND BBS PERIODIC BBS (PBBS) BOOLEAN FORMULAE FOR PBBS PBBS AND NUMERICAL ALGORITHM PBBS AS PERIODIC A(1) M CRYSTAL LATTICE PBBS AS A(1) N?1 CRYSTAL CHAINS FUNDAMENTAL CYCLE OF PBBS TIME IN SCIENCE: REVERSIBILITY VS IRREVERSIBILITY Y POMEAU ON THE PHENOMENON REVERSIBILITY OF RANDOM SIGNALS CONCLUSION AND PERSPECTIVES