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 provides comprehensive reference for researchers already
working in the field.
