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Introduction to Classical Integrable Systems

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Hardcover | 27 June 2003

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Hardcover
616 Pages

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25.4 x 17.78 x 3.33

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Published: 31st May 2007

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This book provides a thorough introduction to the theory of classical integrable systems, discussing the various approaches to the subject and explaining their interrelations. The book begins by introducing the central ideas of the theory of integrable systems, based on Lax representations, loop groups and Riemann surfaces. These ideas are then illustrated with detailed studies of model systems. The connection between isomonodromic deformation and integrability is discussed, and integrable field theories are covered in detail. The KP, KdV and Toda hierarchies are explained using the notion of Grassmannian, vertex operators and pseudo-differential operators. A chapter is devoted to the inverse scattering method and three complementary chapters cover the necessary mathematical tools from symplectic geometry, Riemann surfaces and Lie algebras. The book contains many worked examples and is suitable for use as a textbook on graduate courses. It also provides a comprehensive reference for researchers already working in the field.

Industry Reviews

'This monograph provides a thorough introduction to the theory of classical integrable systems ... The book contains many worked examples and is suitable for use as a textbook on graduate courses. For researchers already working in this field this book is a valuable source of information which provides an excellent overview of the established results and the present developments.' Zentralblatt MATH

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Non-Fiction Science Physics Electricity Engineering & Technology Energy Technology & Engineering Electrical Engineering Mathematics Applied Mathematics Classical Mathematics

Introduction	p. 1
Integrable dynamical systems	p. 5
Introduction	p. 5
The Liouville theorem	p. 7
Action-angle variables	p. 10
Lax pairs	p. 11
Existence of an r-matrix	p. 13
Commuting flows	p. 17
The Kepler problem	p. 17
The Euler top	p. 19
The Lagrange top	p. 20
The Kowalevski top	p. 22
The Neumann model	p. 23
Geodesics on an ellipsoid	p. 25
Separation of variables in the Neumann model	p. 27
Synopsis of integrable systems	p. 32
Examples of Lax pairs with spectral parameter	p. 33
The Zakharov-Shabat construction	p. 35
Coadjoint orbits and Hamiltonian formalism	p. 41
Elementary flows and wave function	p. 49
Factorization problem	p. 54
Tau-functions	p. 59
Integrable field theories and monodromy matrix	p. 62
Abelianization	p. 65
Poisson brackets of the monodromy matrix	p. 72
The group of dressing transformations	p. 74
Soliton solutions	p. 79
Algebraic methods	p. 86
The classical and modified Yang-Baxter equations	p. 86
Algebraic meaning of the classical Yang-Baxter equations	p. 89
Adler-Kostant-Symes scheme	p. 92
Construction of integrable systems	p. 94
Solving by factorization	p. 96
The open Toda chain	p. 97
The r-matrix of the Toda models	p. 100
Solution of the open Toda chain	p. 105
Toda system and Hamiltonian reduction	p. 109
The Lax pair of the Kowalevski top	p. 115
Analytical methods	p. 124
The spectral curve	p. 125
The eigenvector bundle	p. 130
The adjoint linear system	p. 138
Time evolution	p. 142
Theta-functions formulae	p. 145
Baker-Akhiezer functions	p. 149
Linearization and the factorization problem	p. 153
Tau-functions	p. 154
Symplectic form	p. 156
Separation of variables and the spectral curve	p. 162
Action-angle variables	p. 164
Riemann surfaces and integrability	p. 167
The Kowalevski top	p. 169
Infinite-dimensional systems	p. 175
The closed Toda chain	p. 178
The model	p. 178
The spectral curve	p. 181
The eigenvectors	p. 182
Reconstruction formula	p. 184
Symplectic structure	p. 191
The Sklyanin approach	p. 193
The Poisson brackets	p. 196
Reality conditions	p. 200
The Calogero-Moser model	p. 206
The spin Calogero-Moser model	p. 206
Lax pair	p. 208
The r-matrix	p. 210
The scalar Calogero-Moser model	p. 214
The spectral curve	p. 216
The eigenvector bundle	p. 218
Time evolution	p. 220
Reconstruction formulae	p. 221
Symplectic structure	p. 223
Poles systems and double-Bloch condition	p. 226
Hitchin systems	p. 232
Examples of Hitchin systems	p. 239
The trigonometric Calogero-Moser model	p. 244
Isomonodromic deformations	p. 249
Introduction	p. 249
Monodromy data	p. 251
Isomonodromy and the Riemann-Hilbert problem	p. 262
Isomonodromic deformations	p. 264
Schlesinger transformations	p. 270
Tau-functions	p. 272
Ricatti equation	p. 277
Sato's formula	p. 278
The Hirota equations	p. 280
Tau-functions and theta-functions	p. 282
The Painleve equations	p. 290
Grassmannian and integrable hierarchies	p. 299
Introduction	p. 299
Fermions and GL [infinity]	p. 303
Boson-fermion correspondence	p. 308
Tau-functions and Hirota bilinear identities	p. 311
The KP hierarchy and its soliton solutions	p. 314
Fermions and Grassmannians	p. 316
Schur polynomials	p. 322
From fermions to pseudo-differential operators	p. 328
The Segal-Wilson approach	p. 331
The KP hierarchy	p. 338
The algebra of pseudo-differential operators	p. 338
The KP hierarchy	p. 341
The Baker-Akhiezer function of KP	p. 344
Algebro-geometric solutions of KP	p. 348
The tau-function of KP	p. 352
The generalized KdV equations	p. 355
KdV Hamiltonian structures	p. 359
Bihamiltonian structure	p. 363
The Drinfeld-Sokolov reduction	p. 364
Whitham equations	p. 370
Solution of the Whitham equations	p. 379
The KdV hierarchy	p. 382
The KdV equation	p. 382
The KdV hierarchy	p. 386
Hamiltonian structures and Virasoro algebra	p. 392
Soliton solutions	p. 394
Algebro-geometric solutions	p. 398
Finite-zone solutions	p. 408
Action-angle variables	p. 414
Analytical description of solitons	p. 419
Local fields	p. 425
Whitham's equations	p. 433
The Toda field theories	p. 443
The Liouville equation	p. 443
The Toda systems and their zero-curvature representations	p. 445
Solution of the Toda field equations	p. 447
Hamiltonian formalism	p. 454
Conformal structure	p. 456
Dressing transformations	p. 463
The affine sinh-Gordon model	p. 467
Dressing transformations and soliton solutions	p. 471
N-soliton dynamics	p. 474
Finite-zone solutions	p. 481
Classical inverse scattering method	p. 486
The sine-Gordon equation	p. 486
The Jost solutions	p. 487
Inverse scattering as a Riemann-Hilbert problem	p. 496
Time evolution of the scattering data	p. 497
The Gelfand-Levitan-Marchenko equation	p. 498
Soliton solutions	p. 502
Poisson brackets of the scattering data	p. 505
Action-angle variables	p. 510
Symplectic geometry	p. 516
Poisson manifolds and symplectic manifolds	p. 516
Coadjoint orbits	p. 522
Symmetries and Hamiltonian reduction	p. 525
The case M = T*G	p. 532
Poisson-Lie groups	p. 534
Action of a Poisson-Lie group on a symplectic manifold	p. 538
The groups G and G*	p. 540
The group of dressing transformations	p. 542
Riemann surfaces	p. 545
Smooth algebraic curves	p. 545
Hyperelliptic curves	p. 547
The Riemann-Hurwitz formula	p. 549
The field of meromorphic functions of a Riemann surface	p. 549
Line bundles on a Riemann surface	p. 551
Divisors	p. 553
Chern class	p. 554
Serre duality	p. 554
The Riemann-Roch theorem	p. 556
Abelian differentials	p. 559
Riemann bilinear identities	p. 560
Jacobi variety	p. 562
Theta-functions	p. 563
The genus 1 case	p. 567
The Riemann-Hilbert factorization problem	p. 568
Lie algebras	p. 571
Lie groups and Lie algebras	p. 571
Semi-simple Lie algebras	p. 574
Linear representations	p. 580
Real Lie algebras	p. 583
Affine Kac-Moody algebras	p. 587
Vertex operator representations	p. 592
Index	p. 599
Table of Contents provided by Ingram. All Rights Reserved.

Introduction to Classical Integrable Systems

At a Glance

New Edition

Hardcover

Industry Reviews

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