This is an extensive volume devoted to the integrability of nonlinear Hamiltonian differential equations. The book is designed as a teaching textbook and aims at a wide readership of mathematicians and physicists, graduate students and professionals. The main thrust of the book is to show how algebraic geometry, Lie theory and Painlevé analysis can be used to explicitly solve so-called integrable differential equations and to construct the algebraic tori on which they linearize. It is also a playground for students in applying algebraic geometry and Lie theory.
The book is reasonably self-contained and presents numerous examples, which appear through the text to illustrate the ideas and make up the core of the last part of the book. It also provides many fundamental tools from Lie algebras and groups to algebraic and differential geometry, all of which help to understand the main topic-integrability.
The book is divided into three parts. Part I deals with Liouville integrable systems in a real or complex setting of differential geometry. It includes the theory of Poisson manifolds, Lie algebras and Lie groups. Besides classical integrability theories, it also contains contemporary integrability results discovered in the last few decades, for example, Lax operators and \(r\)-matrix theory.
Part II lays the theoretical foundation of algebraic complete integrability. This part contains an outline of algebraic tools for nonspecialists such as Abelian, Jacobi and Prym varieties. Laurent expansions are used to understand the embedding of the complexified invariant manifolds in projective space and their algebraic nature.
To linearize a Hamiltonian system on an Abelian variety, one may either construct a Lax representation of the differential equation involving an extra-parameter and linear on the Jacobian of the corresponding characteristic curve, or one may complete the complexified invariant manifolds by using the Laurent solutions of the differential equation. This latter approach allows one to identify the nature of the invariant manifolds and of the solutions of the system. It is quite usual that the isospectral manifolds and the invariant manifolds are different.
Part III exhibits three classes of examples of completely integrable systems. The first class contains integrable geodesic flow on \(SO(4)\). The second class addresses the classification of integrable lattices among a natural class of Toda-like lattices. The last class of examples show the classification of rigid body motions about a fixed-point under the influence of gravity. The presented examples contain the Euler, Lagrange, Kowalewski and Goryachev-Chaplygin tops.
The book provides many useful tools and techniques in the field of completely integrable systems. It is a valuable source for graduate students and researchers who like to enter the integrability theory or to learn fascinating aspects of integrable geometry of nonlinear differential equations.