Painlevé Equations through Symmetry
About this Title
Masatoshi Noumi, Kobe University, Kobe, Japan. Translated by Professor Masatoshi Noumi
Publication: Translations of Mathematical Monographs
Publication Year: 2004; Volume 223
ISBNs: 978-0-8218-3221-9 (print); 978-1-4704-4647-5 (online)
MathSciNet review: MR2044201
MSC: Primary 34M15; Secondary 05E05, 05E10, 33E17, 34C14, 34M55, 37J35, 39A12
The six Painlevé equations (nonlinear ordinary differential equations of the second order with nonmovable singularities) have attracted the attention of mathematicians for more than a hundred years. These equations and their solutions (Painlevé transcendents) nowadays play an important role in many areas of mathematics, such as the theory of special functions, the theory of integrable systems, differential geometry, and mathematical aspects of quantum field theory.
The present book is devoted to one of the aspects of the theory of Painlevé equations, namely to their symmetry properties. For several types of Painlevé equations (especially equations of types II and IV), the author studies families of transformations—the so-called Bäcklund transformations—which transform solutions of a given Painlevé equations to solutions of the same equations with a different set of parameters. It turns out that these symmetries can be interpreted in terms of root systems associated to affine Weyl groups. The author describes remarkable combinatorial structures of these symmetries and shows how they are related to the theory of $\tau$-functions associated to integrable systems.
Graduate students and research mathematicians interested in special functions and the theory of integrable systems.
Table of Contents
- What is a Bäcklund transformation?
- The symmetric form
- $\tau $-functions
- $\tau $-functions on the lattice
- Jacobi-Trudi formula
- Getting familiar with determinants
- Gauss decomposition and birational transformations
- Lax formalism