**Translations of Mathematical Monographs**

2004;
156 pp;
Hardcover

MSC: Primary 34;
Secondary 37; 39; 14; 20

Print ISBN: 978-0-8218-3221-9

Product Code: MMONO/223

List Price: $73.00

AMS Member Price: $58.40

MAA Member Price: $65.70

**Electronic ISBN: 978-1-4704-4647-5
Product Code: MMONO/223.E**

List Price: $73.00

AMS Member Price: $58.40

MAA Member Price: $65.70

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#### Supplemental Materials

# Painlevé Equations through Symmetry

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*Masatoshi Noumi*

“The Painlevé equations themselves are really a wonder. They still continue to give us fresh mysteries … One reason that I wrote this book is to tell you how impressed I am by the mysteries of the Painlevé equations.”

—from the Preface

The six Painlevé equations (nonlinear ordinary differential equations
of the second order with nonmovable singularities) have attracted the attention
of mathematicians for more than 100 years. These equations and their solutions,
the 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 the symmetry of Painlevé equations
(especially those of types II and IV). The author studies families of
transformations for several types of Painlevé equations—the
so-called Bäcklund transformations—which transform solutions of a
given Painlevé equation to solutions of the same equation 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 the remarkable combinatorial structures of these
symmetries, and shows how they are related to the theory of
\(\tau\)-functions associated to integrable systems.

Prerequisites include undergraduate calculus and linear algebra with some
knowledge of group theory. The book is suitable for graduate students and
research mathematicians interested in special functions and the theory of
integrable systems.

#### Readership

Graduate students and research mathematicians interested in special functions and the theory of integrable systems.

#### Reviews & Endorsements

This book provides a new perspective on these materials, and is recommended to those who are interested in this field.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Painleve Equations through Symmetry

- Cover Cover11
- Title page i2
- Contents iii4
- Preface v6
- Preface to the English edition ix10
- What is a Bäcklund transformation? 112
- The symmetric form 1324
- 𝜏-functions 2738
- 𝜏-functions on the lattice 4556
- Jacobi-Trudi formula 6778
- Getting familiar with determinants 8596
- Gauss decomposition and birational transformations 99110
- Lax formalism 125136
- Appendix 133144
- Bibliography 153164
- Index 155166
- Back Cover Back Cover1170