Introduction to classical integrable systems / Olivier Babelon, Denis Bernard, Michel Talon.

Format:

Book

Author/Creator:

Babelon, Olivier, 1951-

Contributor:

Bernard, Denis, 1961-

Talon, Michel, 1952-

Rosengarten Family Fund.

Series:

Cambridge monographs on mathematical physics

Language:

English

Subjects (All):

Dynamics.

Hamiltonian systems.

Physical Description:

xi, 602 pages : illustrations ; 26 cm.

Place of Publication:

Cambridge ; New York : Cambridge University Press, 2003.

Summary:

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.

Contents:

2 Integrable dynamical systems 5

2.2 The Liouville theorem 7

2.3 Action-angle variables 10

2.4 Lax pairs 11

2.5 Existence of an r-matrix 13

2.6 Commuting flows 17

2.7 The Kepler problem 17

2.8 The Euler top 19

2.9 The Lagrange top 20

2.10 The Kowalevski top 22

2.11 The Neumann model 23

2.12 Geodesics on an ellipsoid 25

2.13 Separation of variables in the Neumann model 27

3 Synopsis of integrable systems 32

3.1 Examples of Lax pairs with spectral parameter 33

3.2 The Zakharov-Shabat construction 35

3.3 Coadjoint orbits and Hamiltonian formalism 41

3.4 Elementary flows and wave function 49

3.5 Factorization problem 54

3.6 Tau-functions 59

3.7 Integrable field theories and monodromy matrix 62

3.8 Abelianization 65

3.9 Poisson brackets of the monodromy matrix 72

3.10 The group of dressing transformations 74

3.11 Soliton solutions 79

4 Algebraic methods 86

4.1 The classical and modified Yang-Baxter equations 86

4.2 Algebraic meaning of the classical Yang-Baxter equations 89

4.3 Adler-Kostant-Symes scheme 92

4.4 Construction of integrable systems 94

4.5 Solving by factorization 96

4.6 The open Toda chain 97

4.7 The r-matrix of the Toda models 100

4.8 Solution of the open Toda chain 105

4.9 Toda system and Hamiltonian reduction 109

4.10 The Lax pair of the Kowalevski top 115

5 Analytical methods 124

5.1 The spectral curve 125

5.2 The eigenvector bundle 130

5.3 The adjoint linear system 138

5.4 Time evolution 142

5.5 Theta-functions formulae 145

5.6 Baker-Akhiezer functions 149

5.7 Linearization and the factorization problem 153

5.8 Tau-functions 154

5.9 Symplectic form 156

5.10 Separation of variables and the spectral curve 162

5.11 Action-angle variables 164

5.12 Riemann surfaces and integrability 167

5.13 The Kowalevski top 169

5.14 Infinite-dimensional systems 175

6 The closed Toda chain 178

6.1 The model 178

6.2 The spectral curve 181

6.3 The eigenvectors 182

6.4 Reconstruction formula 184

6.5 Symplectic structure 191

6.6 The Sklyanin approach 193

6.7 The Poisson brackets 196

6.8 Reality conditions 200

7 The Calogero-Moser model 206

7.1 The spin Calogero-Moser model 206

7.2 Lax pair 208

7.3 The r-matrix 210

7.4 The scalar Calogero-Moser model 214

7.5 The spectral curve 216

7.6 The eigenvector bundle 218

7.7 Time evolution 220

7.8 Reconstruction formulae 221

7.9 Symplectic structure 223

7.10 Poles systems and double-Bloch condition 226

7.11 Hitchin systems 232

7.12 Examples of Hitchin systems 239

7.13 The trigonometric Calogero-Moser model 244

8 Isomonodromic deformations 249

8.2 Monodromy data 251

8.3 Isomonodromy and the Riemann-Hilbert problem 262

8.4 Isomonodromic deformations 264

8.5 Schlesinger transformations 270

8.6 Tau-functions 272

8.7 Ricatti equation 277

8.8 Sato's formula 278

8.9 The Hirota equations 280

8.10 Tau-functions and theta-functions 282

8.11 The Painleve equations 290

9 Grassmannian and integrable hierarchies 299

9.2 Fermions and GL [infinity] 303

9.3 Boson-fermion correspondence 308

9.4 Tau-functions and Hirota bilinear identities 311

9.5 The KP hierarchy and its soliton solutions 314

9.6 Fermions and Grassmannians 316

9.7 Schur polynomials 322

9.8 From fermions to pseudo-differential operators 328

9.9 The Segal-Wilson approach 331

10 The KP hierarchy 338

10.1 The algebra of pseudo-differential operators 338

10.2 The KP hierarchy 341

10.3 The Baker-Akhiezer function of KP 344

10.4 Algebro-geometric solutions of KP 348

10.5 The tau-function of KP 352

10.6 The generalized KdV equations 355

10.7 KdV Hamiltonian structures 359

10.8 Bihamiltonian structure 363

10.9 The Drinfeld-Sokolov reduction 364

10.10 Whitham equations 370

10.11 Solution of the Whitham equations 379

11 The KdV hierarchy 382

11.1 The KdV equation 382

11.2 The KdV hierarchy 386

11.3 Hamiltonian structures and Virasoro algebra 392

11.4 Soliton solutions 394

11.5 Algebro-geometric solutions 398

11.6 Finite-zone solutions 408

11.7 Action-angle variables 414

11.8 Analytical description of solitons 419

11.9 Local fields 425

11.10 Whitham's equations 433

12 The Toda field theories 443

12.1 The Liouville equation 443

12.2 The Toda systems and their zero-curvature representations 445

12.3 Solution of the Toda field equations 447

12.4 Hamiltonian formalism 454

12.5 Conformal structure 456

12.6 Dressing transformations 463

12.7 The affine sinh-Gordon model 467

12.8 Dressing transformations and soliton solutions 471

12.9 N-soliton dynamics 474

12.10 Finite-zone solutions 481

13 Classical inverse scattering method 486

13.1 The sine-Gordon equation 486

13.2 The Jost solutions 487

13.3 Inverse scattering as a Riemann-Hilbert problem 496

13.4 Time evolution of the scattering data 497

13.5 The Gelfand-Levitan-Marchenko equation 498

13.6 Soliton solutions 502

13.7 Poisson brackets of the scattering data 505

13.8 Action-angle variables 510

14 Symplectic geometry 516

14.1 Poisson manifolds and symplectic manifolds 516

14.2 Coadjoint orbits 522

14.3 Symmetries and Hamiltonian reduction 525

14.4 The case M = T*G 532

14.5 Poisson-Lie groups 534

14.6 Action of a Poisson-Lie group on a symplectic manifold 538

14.7 The groups G and G* 540

14.8 The group of dressing transformations 542

15 Riemann surfaces 545

15.1 Smooth algebraic curves 545

15.2 Hyperelliptic curves 547

15.3 The Riemann-Hurwitz formula 549

15.4 The field of meromorphic functions of a Riemann surface 549

15.5 Line bundles on a Riemann surface 551

15.6 Divisors 553

15.7 Chern class 554

15.8 Serre duality 554

15.9 The Riemann-Roch theorem 556

15.10 Abelian differentials 559

15.11 Riemann bilinear identities 560

15.12 Jacobi variety 562

15.13 Theta-functions 563

15.14 The genus 1 case 567

15.15 The Riemann-Hilbert factorization problem 568

16 Lie algebras 571

16.1 Lie groups and Lie algebras 571

16.2 Semi-simple Lie algebras 574

16.3 Linear representations 580

16.4 Real Lie algebras 583

16.5 Affine Kac-Moody algebras 587

16.6 Vertex operator representations 592.

Notes:

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Rosengarten Family Fund.

ISBN:

052182267X

OCLC:

50693556

Online:

Publisher description