Introduction to Classical Integrable Systems

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Cambridge University Press, Apr 17, 2003 - Science - 602 pages
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.

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2 Integrable dynamical systems
3 Synopsis of integrable systems
4 Algebraic methods
5 Analytical methods
6 The closed Toda chain
7 The CalogeroMoser model
8 Isomonodromic deformations
9 Grassmannian and integrable hierarchies
11 The KdV hierarchy
12 The Toda field theories
13 Classical inverse scattering method
14 Symplectic geometry
15 Riemann surfaces
16 Lie algebras

10 The KP hierarchy

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About the author (2003)

Olivier Babelon has been a member of the Centre National de la Recherche Scientifique (CNRS) since 1978. He works at the Laboratoire de Physique Théorique et Hautes Energies (LPTHE) at the University of Paris VI-Paris VII. His main fields of interest are particle physics, gauge theories and integrables systems.

Denis Bernard has been a member of the CNRS since 1988. He currently works at the Service de Physique Théorique de Saclay. His main fields of interest are conformal field theories and integrable systems, and other aspects of statistical field theories, including statistical turbulence.

Michel Talon has been a member of the CNRS since 1977. He works at the LPTHE at the University of Paris VI-Paris VII. He is involved in the computation of radiative corrections and anomalies in gauge theories and integrable systems.

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