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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