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Painlevé Equations through Symmetry
 
Masatoshi Noumi Kobe University, Kobe, Japan
Painleve Equations through Symmetry
eBook ISBN:  978-1-4704-4647-5
Product Code:  MMONO/223.E
List Price: $155.00
MAA Member Price: $139.50
AMS Member Price: $124.00
Painleve Equations through Symmetry
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Painlevé Equations through Symmetry
Masatoshi Noumi Kobe University, Kobe, Japan
eBook ISBN:  978-1-4704-4647-5
Product Code:  MMONO/223.E
List Price: $155.00
MAA Member Price: $139.50
AMS Member Price: $124.00
  • Book Details
     
     
    Translations of Mathematical Monographs
    Volume: 2232004; 156 pp
    MSC: Primary 34; Secondary 37; 39; 14; 20;

    “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.

  • Table of Contents
     
     
    • Chapters
    • 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
  • Additional Material
     
     
  • Reviews
     
     
    • This book provides a new perspective on these materials, and is recommended to those who are interested in this field.

      Zentralblatt MATH
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2232004; 156 pp
MSC: Primary 34; Secondary 37; 39; 14; 20;

“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.

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

    Zentralblatt MATH
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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