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O. Babelon,D. Bernard, & M. Talon, Introduction to Classical Integrable Systems

'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.'

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.

Discusses all approaches to the subject and emphasises their unity

Written in a clear and pedagogical style

Includes developments of importance for the study of quantum systems

Contents

1. Introduction; 2. Integrable dynamical systems; 3. Synopsis of integrable systems; 4. Algebraic methods;

5. Analytical methods; 6. The closed Toda chain; 7. The Calogero-Moser model; 8. Isomonodromic deformations;

9. Grassmannian and integrable hierarchies; 10. The KP hierarchy; 11. The KdV hierarchy;

12. The Toda field theories; 13. Classical inverse scattering method; 14. Symplectic geometry;

15. Riemann surfaces; 16. Lie algebras; Index.

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