ELEMENTARY TREATISE ON PHYSICS EXPERIMENTAL AND APPLIED

By E. NASII

CONTENTS INTRODUCTION INTEGRABLE DYNAMICAL SYSTEMS THE LIOUVILLE THEOREM ACTION–ANGLE VARIABLES LAX PAIRS EXISTENCE OF AN R-MATRIX COMMUTING ?OWS THE KEPLER PROBLEM THE EULER TOP THE LAGRANGE TOP THE KOWALEVSKI TOP THE NEUMANN MODEL GEODESICS ON AN ELLIPSOID SEPARATION OF VARIABLES IN THE NEUMANN MODEL SYNOPSIS OF INTEGRABLE SYSTEMS EXAMPLES OF LAX PAIRS WITH SPECTRAL PARAMETER THE ZAKHAROV–SHABAT CONSTRUCTION COADJOINT ORBITS AND HAMILTONIAN FORMALISM ELEMENTARY ?OWS AND WAVE FUNCTION FACTORIZATION PROBLEM TAU-FUNCTIONS INTEGRABLE ?ELD THEORIES AND MONODROMY MATRIX ABELIANIZATION POISSON BRACKETS OF THE MONODROMY MATRIX THE GROUP OF DRESSING TRANSFORMATIONS SOLITON SOLUTIONS ALGEBRAIC METHODS THE CLASSICAL AND MODI?ED YANG–BAXTER EQUATIONS ALGEBRAIC MEANING OF THE CLASSICAL YANG–BAXTER EQUATIONS ADLER–KOSTANT–SYMES SCHEME CONSTRUCTION OF INTEGRABLE SYSTEMS SOLVING BY FACTORIZATION THE OPEN TODA CHAIN THE R-MATRIX OF THE TODA MODELS SOLUTION OF THE OPEN TODA CHAIN TODA SYSTEM AND HAMILTONIAN REDUCTION THE LAX PAIR OF THE KOWALEVSKI TOP ANALYTICAL METHODS THE SPECTRAL CURVE THE EIGENVECTOR BUNDLE THE ADJOINT LINEAR SYSTEM TIME EVOLUTION THETA-FUNCTIONS FORMULAE BAKER–AKHIEZER FUNCTIONS LINEARIZATION AND THE FACTORIZATION PROBLEM SYMPLECTIC FORM SEPARATION OF VARIABLES AND THE SPECTRAL CURVE RIEMANN SURFACES AND INTEGRABILITY IN?NITE-DIMENSIONAL SYSTEMS THE CLOSED TODA CHAIN THE MODEL THE EIGENVECTORS RECONSTRUCTION FORMULA SYMPLECTIC STRUCTURE THE SKLYANIN APPROACH THE POISSON BRACKETS REALITY CONDITIONS THE CALOGERO–MOSER MODEL THE SPIN CALOGERO–MOSER MODEL LAX PAIR THE R-MATRIX THE SCALAR CALOGERO–MOSER MODEL RECONSTRUCTION FORMULAE POLES SYSTEMS AND DOUBLE-BLOCH CONDITION HITCHIN SYSTEMS EXAMPLES OF HITCHIN SYSTEMS THE TRIGONOMETRIC CALOGERO–MOSER MODEL ISOMONODROMIC DEFORMATIONS MONODROMY DATA ISOMONODROMY AND THE RIEMANN–HILBERT PROBLEM SCHLESINGER TRANSFORMATIONS RICATTI EQUATION SATO’S FORMULA THE HIROTA EQUATIONS TAU-FUNCTIONS AND THETA-FUNCTIONS THE PAINLEV´E EQUATIONS GRASSMANNIAN AND INTEGRABLE HIERARCHIES FERMIONS AND GL(?) BOSON–FERMION CORRESPONDENCE TAU-FUNCTIONS AND HIROTA BILINEAR IDENTITIES THE KP HIERARCHY AND ITS SOLITON SOLUTIONS FERMIONS AND GRASSMANNIANS SCHUR POLYNOMIALS FROM FERMIONS TO PSEUDO-DI?ERENTIAL OPERATORS THE SEGAL–WILSON APPROACH THE KP HIERARCHY THE ALGEBRA OF PSEUDO-DI?ERENTIAL OPERATORS THE BAKER–AKHIEZER FUNCTION OF KP ALGEBRO-GEOMETRIC SOLUTIONS OF KP THE TAU-FUNCTION OF KP THE GENERALIZED KDV EQUATIONS KDV HAMILTONIAN STRUCTURES BIHAMILTONIAN STRUCTURE THE DRINFELD–SOKOLOV REDUCTION 10 10 WHITHAM EQUATIONS 10 11 SOLUTION OF THE WHITHAM EQUATIONS THE KDV HIERARCHY THE KDV EQUATION HAMILTONIAN STRUCTURES AND VIRASORO ALGEBRA ALGEBRO-GEOMETRIC SOLUTIONS FINITE-ZONE SOLUTIONS ACTION-ANGLE VARIABLES ANALYTICAL DESCRIPTION OF SOLITONS LOCAL ?ELDS 11 10 WHITHAM’S EQUATIONS THE TODA ?ELD THEORIES THE LIOUVILLE EQUATION THE TODA SYSTEMS AND THEIR ZERO-CURVATURE REPRESENTATIONS SOLUTION OF THE TODA ?ELD EQUATIONS HAMILTONIAN FORMALISM CONFORMAL STRUCTURE DRESSING TRANSFORMATIONS THE A?NE SINH-GORDON MODEL DRESSING TRANSFORMATIONS AND SOLITON SOLUTIONS N-SOLITON DYNAMICS 12 10 FINITE-ZONE SOLUTIONS CLASSICAL INVERSE SCATTERING METHOD THE SINE-GORDON EQUATION THE JOST SOLUTIONS INVERSE SCATTERING AS A RIEMANN--HILBERT PROBLEM TIME EVOLUTION OF THE SCATTERING DATA THE GELFAND--LEVITAN--MARCHENKO EQUATION POISSON BRACKETS OF THE SCATTERING DATA ACTION--ANGLE VARIABLES SYMPLECTIC GEOMETRY POISSON MANIFOLDS AND SYMPLECTIC MANIFOLDS COADJOINT ORBITS SYMMETRIES AND HAMILTONIAN REDUCTION THE CASE M = T ?G POISSON–LIE GROUPS ACTION OF A POISSON–LIE GROUP ON A SYMPLECTIC MANIFOLD THE GROUPS G AND G? RIEMANN SURFACES SMOOTH ALGEBRAIC CURVES HYPERELLIPTIC CURVES THE RIEMANN–HURWITZ FORMULA THE ?ELD OF MEROMORPHIC FUNCTIONS OF A RIEMANN SURFACE LINE BUNDLES ON A RIEMANN SURFACE DIVISORS CHERN CLASS SERRE DUALITY THE RIEMANN–ROCH THEOREM 15 10 ABELIAN DI?ERENTIALS 15 11 RIEMANN BILINEAR IDENTITIES 15 12 JACOBI VARIETY 15 13 THETA-FUNCTIONS 15 14 THE GENUS 1 CASE 15 15 THE RIEMANN–HILBERT FACTORIZATION PROBLEM LIE ALGEBRAS LIE GROUPS AND LIE ALGEBRAS SEMI-SIMPLE LIE ALGEBRAS LINEAR REPRESENTATIONS REAL LIE ALGEBRAS A?NE KAC–MOODY ALGEBRAS VERTEX OPERATOR REPRESENTATIONS