Introduction to Classical Integrable SystemsThis 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
| 1 | |
| 5 | |
3 Synopsis of integrable systems | 32 |
4 Algebraic methods | 86 |
5 Analytical methods | 124 |
6 The closed Toda chain | 178 |
7 The CalogeroMoser model | 206 |
8 Isomonodromic deformations | 249 |
10 The KP hierarchy | 338 |
11 The KdV hierarchy | 382 |
12 The Toda field theories | 443 |
13 Classical inverse scattering method | 486 |
14 Symplectic geometry | 516 |
15 Riemann surfaces | 545 |
16 Lie algebras | 571 |
| 599 | |
Other editions - View all
Introduction to Classical Integrable Systems Olivier Babelon,Denis Bernard,Michel Talon Limited preview - 2003 |
Introduction to Classical Integrable Systems Olivier Babelon,Denis Bernard,Michel Talon No preview available - 2007 |
Common terms and phrases
Abelian action algebra G analytic asymptotic Baker-Akhiezer function bilinear branch points Cartan Chapter coadjoint coadjoint action coadjoint orbit coefficients components compute condition conserved quantities consider constant coordinates defined definition denote diagonal matrix differential equation differential operator eigenvalues eigenvector equations of motion expression fermionic field finite formula Hamiltonian Hence highest weight holomorphic identity integrable systems introduce invariant KdV hierarchy Lax equation Lax matrix Lax pair Lie algebra Lie group line bundle linear Liouville loop algebra meromorphic function monodromy monodromy matrix normalized orbit parameters phase space Poisson brackets poles polynomial Proof Proposition r-matrix relation representation residue Riemann surface simple singular soliton solution spectral curve structure subalgebra symplectic form tau-function theorem theta-functions tion vanishes variables vector write zeroes αλ