Introducing the reader to classical integrable systems and their applications, this book synthesizes the different approaches to the subject, providing a set of interconnected methods for solving problems in mathematical physics. The authors introduce and explain each method, and demonstrate how it can be applied to particular examples. Rather than presenting an exhaustive list of the various integrable systems, they focus on classical objects which have well-known quantum counterparts, or are the semi-classical limits of quantum objects. They thus enable readers to understand the literature on quantum integrable systems.
Author(s): Olivier Babelon, Denis Bernard, Michel Talon
Series: Cambridge Monographs on Mathematical Physics
Edition: 1
Publisher: Cambridge University Press
Year: 2007
Language: English
Pages: 616
Cover......Page 1
Half-title......Page 3
Series-title......Page 5
Title......Page 7
Copyright......Page 8
Contents......Page 9
1 Introduction......Page 15
2.1 Introduction......Page 19
2.2 The Liouville theorem......Page 21
2.3 Action–angle variables......Page 24
2.4 Lax pairs......Page 25
2.5 Existence of an r-matrix......Page 27
2.7 The Kepler problem......Page 31
2.8 The Euler top......Page 33
2.9 The Lagrange top......Page 34
2.10 The Kowalevski top......Page 36
2.11 The Neumann model......Page 37
2.12 Geodesics on an ellipsoid......Page 39
2.13 Separation of variables in the Neumann model......Page 41
References......Page 44
3 Synopsis of integrable systems......Page 46
3.1 Examples of Lax pairs with spectral parameter......Page 47
3.2 The Zakharov–Shabat construction......Page 49
3.3 Coadjoint orbits and Hamiltonian formalism......Page 55
3.4 Elementary flows and wave function......Page 63
3.5 Factorization problem......Page 68
3.6 Tau-functions......Page 73
3.7 Integrable field theories and monodromy matrix......Page 76
3.8 Abelianization......Page 79
3.9 Poisson brackets of the monodromy matrix......Page 86
3.10 The group of dressing transformations......Page 88
3.11 Soliton solutions......Page 93
References......Page 99
4.1 The classical and modified Yang–Baxter equations......Page 100
4.2 Algebraic meaning of the classical Yang–Baxter equations......Page 103
4.3 Adler–Kostant–Symes scheme......Page 106
4.4 Construction of integrable systems......Page 108
4.5 Solving by factorization......Page 110
4.6 The open Toda chain......Page 111
4.7 The r-matrix of the Toda models......Page 114
4.8 Solution of the open Toda chain......Page 119
4.9 Toda system and Hamiltonian reduction......Page 123
4.10 The Lax pair of the Kowalevski top......Page 129
References......Page 137
5 Analytical methods......Page 138
5.1 The spectral curve......Page 139
5.2 The eigenvector bundle......Page 144
5.3 The adjoint linear system......Page 152
5.4 Time evolution......Page 156
5.5 Theta-functions formulae......Page 159
5.6 Baker–Akhiezer functions......Page 163
5.7 Linearization and the factorization problem......Page 167
5.8 Tau-functions......Page 168
5.9 Symplectic form......Page 170
5.10 Separation of variables and the spectral curve......Page 176
5.11 Action–angle variables......Page 178
5.12 Riemann surfaces and integrability......Page 181
5.13 The Kowalevski top......Page 183
5.14 Infinite-dimensional systems......Page 189
References......Page 190
6.1 The model......Page 192
6.2 The spectral curve......Page 195
6.3 The eigenvectors......Page 196
6.4 Reconstruction formula......Page 198
6.5 Symplectic structure......Page 205
6.6 The Sklyanin approach......Page 207
6.7 The Poisson brackets......Page 210
6.8 Reality conditions......Page 214
References......Page 219
7.1 The spin Calogero–Moser model......Page 220
7.2 Lax pair......Page 222
7.3 The r-matrix......Page 224
7.4 The scalar Calogero–Moser model......Page 228
7.5 The spectral curve......Page 230
7.6 The eigenvector bundle......Page 232
7.7 Time evolution......Page 234
7.8 Reconstruction formulae......Page 235
7.9 Symplectic structure......Page 237
7.10 Poles systems and double-Bloch condition......Page 240
7.11 Hitchin systems......Page 246
7.12 Examples of Hitchin systems......Page 253
7.13 The trigonometric Calogero–Moser model......Page 258
References......Page 261
8.1 Introduction......Page 263
8.2 Monodromy data......Page 265
8.3 Isomonodromy and the Riemann–Hilbert problem......Page 276
8.4 Isomonodromic deformations......Page 278
8.5 Schlesinger transformations......Page 284
8.6 Tau-functions......Page 286
8.7 Ricatti equation......Page 291
8.8 Sato’s formula......Page 292
8.9 The Hirota equations......Page 294
8.10 Tau-functions and theta-functions......Page 296
8.11 The Painlevé equations......Page 304
References......Page 311
9.1 Introduction......Page 313
9.2 Fermions…......Page 317
9.3 Boson–fermion correspondence......Page 322
9.4 Tau-functions and Hirota bilinear identities......Page 325
9.5 The KP hierarchy and its soliton solutions......Page 328
9.6 Fermions and Grassmannians......Page 330
9.7 Schur polynomials......Page 336
9.8 From fermions to pseudo-differential operators......Page 342
9.9 The Segal–Wilson approach......Page 345
References......Page 351
10.1 The algebra of pseudo-differential operators......Page 352
10.2 The KP hierarchy......Page 355
10.3 The Baker–Akhiezer function of KP......Page 358
10.4 Algebro-geometric solutions of KP......Page 362
10.5 The tau-function of KP......Page 366
10.6 The generalized KdV equations......Page 369
10.7 KdV Hamiltonian structures......Page 373
10.8 Bihamiltonian structure......Page 377
10.9 The Drinfeld–Sokolov reduction......Page 378
10.10 Whitham equations......Page 384
10.11 Solution of the Whitham equations......Page 393
References......Page 394
11.1 The KdV equation......Page 396
11.2 The KdV hierarchy......Page 400
11.3 Hamiltonian structures and Virasoro algebra......Page 406
11.4 Soliton solutions......Page 408
11.5 Algebro-geometric solutions......Page 412
11.6 Finite-zone solutions......Page 422
11.7 Action-angle variables......Page 428
11.8 Analytical description of solitons......Page 433
11.9 Local fields......Page 439
11.10 Whitham’s equations......Page 447
References......Page 455
12.1 The Liouville equation......Page 457
12.2 The Toda systems and their zero-curvature representations......Page 459
12.3 Solution of the Toda field equations......Page 461
12.4 Hamiltonian formalism......Page 468
12.5 Conformal structure......Page 470
12.6 Dressing transformations......Page 477
12.7 The affine sinh-Gordon model......Page 481
12.8 Dressing transformations and soliton solutions......Page 485
12.9 N-soliton dynamics......Page 488
12.10 Finite-zone solutions......Page 495
References......Page 498
13.1 The sine-Gordon equation......Page 500
13.2 The Jost solutions......Page 501
13.3 Inverse scattering as a Riemann–Hilbert problem......Page 510
13.4 Time evolution of the scattering data......Page 511
13.5 The Gelfand–Levitan–Marchenko equation......Page 512
13.6 Soliton solutions......Page 516
13.7 Poisson brackets of the scattering data......Page 519
13.8 Action–angle variables......Page 524
References......Page 529
14.1 Poisson manifolds and symplectic manifolds......Page 530
14.2 Coadjoint orbits......Page 536
14.3 Symmetries and Hamiltonian reduction......Page 539
14.4 The case M = T G......Page 546
14.5 Poisson–Lie groups......Page 548
14.6 Action of a Poisson–Lie group on a symplectic manifold......Page 552
14.7 The groups G and G......Page 554
14.8 The group of dressing transformations......Page 556
References......Page 557
15.1 Smooth algebraic curves......Page 559
15.2 Hyperelliptic curves......Page 561
15.4 The field of meromorphic functions of a Riemann surface......Page 563
15.5 Line bundles on a Riemann surface......Page 565
15.6 Divisors......Page 567
15.8 Serre duality......Page 568
15.9 The Riemann–Roch theorem......Page 570
15.10 Abelian differentials......Page 573
15.11 Riemann bilinear identities......Page 574
15.12 Jacobi variety......Page 576
15.13 Theta-functions......Page 577
15.14 The genus 1 case......Page 581
15.15 The Riemann–Hilbert factorization problem......Page 582
References......Page 584
16.1 Lie groups and Lie algebras......Page 585
16.2 Semi-simple Lie algebras......Page 588
16.3 Linear representations......Page 594
16.4 Real Lie algebras......Page 597
16.5 Affine Kac–Moody algebras......Page 601
16.6 Vertex operator representations......Page 606
References......Page 611
Index......Page 613
