by L A Dickey (University of Oklahoma)

The theory of soliton equations and integrable systems has developed rapidly during the last 20 years with numerous applications in mechanics and physics. For a long time books in this field have not been written but the flood of papers was overwhelming: many hundreds, maybe thousands of them. All this followed one single work by Gardner, Greene, Kruskal, and Miura about the Korteweg-de Vries equation (KdV) which, had seemed to be merely an unassuming equation of mathematical physics describing waves in shallow water.

This branch of science is attractive because it is one of those which revives the interest in the basic principles of mathematics, a beautiful formula.

• Integrable Systems Generated by Linear Differential nth Order Operators
• Hamiltonian Structures
• Hamiltonian Structures of the KdV-Hierarchies
• The Kupershmidt-Wilson Theorem
• The KP-Hierarchy
• Hamiltonian Structure of the KP-Hierarchy
• Baker Function, τ-Function
• Grassmannian, τ-Function and Baker Function after Segal and Wilson. Algebraic-Geometrical Krichever's Solutions
• Matrix First-Order Operators
• KdV-Hierarchies as Reductions of Matrix Hierarchies
• Stationary Equations
• Stationary Equations of the KdV-Hierarchy in the Narrow Sense (n=2)
• Stationary Equations of the Matrix Hierarchy
• Stationary Equations of the KdV-Hierarchies
• Matrix Differential Operators Polynomially Depending on a Parameter
• Multi-time Lagrangian and Hamiltonian Formalism
• Further Examples and Applications