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Padé Methods for Painlevé Equations

  • Book
  • © 2021

Overview

Authors:
  1. Hidehito Nagao

    1. Natural Sciences Division, National Institute of Technology, Akashi College, Akashi, Japan

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  2. Yasuhiko Yamada

    1. Department of Mathematics, Kobe University, Kobe, Japan

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    You can also search for this author in PubMed   Google Scholar

  • Presents an elemental method, assuming only standard linear algebra and complex analysis
  • Allows target equations such as Painlevé and Garnier systems to arise naturally through suitable Padé problems
  • Provides a unique guide to continuous and discrete isomonodromic deformation equations based on a simple method

Part of the book series: SpringerBriefs in Mathematical Physics (BRIEFSMAPHY, volume 42)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-viii
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  2. Padé Approximation and Differential Equation

    • Hidehtio Nagao, Yasuhiko Yamada
    Pages 1-8

  3. Padé Approximation for \(P_\mathrm{VI}\)

    • Hidehtio Nagao, Yasuhiko Yamada
    Pages 9-30

  4. Padé Approximation for q-Painlevé/Garnier Equations

    • Hidehtio Nagao, Yasuhiko Yamada
    Pages 31-47

  5. Padé Interpolation

    • Hidehtio Nagao, Yasuhiko Yamada
    Pages 49-59

  6. Padé Interpolation on q-Quadratic Grid

    • Hidehtio Nagao, Yasuhiko Yamada
    Pages 61-73

  7. Multicomponent Generalizations

    • Hidehtio Nagao, Yasuhiko Yamada
    Pages 75-83

  8. Back Matter

    Pages 85-90
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Keywords

About this book

The isomonodromic deformation equations such as the Painlevé and Garnier systems are an important class of nonlinear differential equations in mathematics and mathematical physics. For discrete analogs of these equations in particular, much progress has been made in recent decades. Various approaches to such isomonodromic equations are known: the Painlevé test/Painlevé property, reduction of integrable hierarchy, the Lax formulation, algebro-geometric methods, and others. Among them, the Padé method explained in this book provides a simple approach to those equations in both continuous and discrete cases.


For a given function f(x), the Padé approximation/interpolation supplies the rational functions P(x), Q(x) as approximants such as f(x)~P(x)/Q(x). The basic idea of the Padé method is to consider the linear differential (or difference) equations satisfied by P(x) and f(x)Q(x). In choosing the suitable approximation problem, the linear differential equations give the Lax pair for some isomonodromic equations. Although this relation between the isomonodromic equations and Padé approximations has been known classically, a systematic study including discrete cases has been conducted only recently. By this simple and easy procedure, one can simultaneously obtain various results such as the nonlinear evolution equation, its Lax pair, and their special solutions. In this way, the method is a convenient means of approaching the isomonodromic deformation equations.

Reviews

“The monograph under review explores an important connection between integrable systems and approximation theory: the appearance of integrable systems in Padé approximation and interpolation problems. Both authors have contributed plenty of work in this direction together and individually, and one can view this work as a pedagogical guide to this rich area. … The monograph is well organized, and has plenty of examples motivating the discussion and demonstrating the power of the Padé method … .” (Ahmad Bassam Barhoumi, Mathematical Reviews, September, 2022)

Authors and Affiliations

  • Natural Sciences Division, National Institute of Technology, Akashi College, Akashi, Japan

    Hidehito Nagao

  • Department of Mathematics, Kobe University, Kobe, Japan

    Yasuhiko Yamada

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