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Painlevé Transcendents: The Riemann-Hilbert Approach

Publisher: 
American Mathematical Society
Number of Pages: 
550
Price: 
109.00
ISBN: 
082183651X
Date Received: 
Tuesday, October 24, 2006
Reviewable: 
No
Include In BLL Rating: 
No
Reviewer Email Address: 
Athanassios S. Fokas, Alexander R. Its, Andrei A. Kapaev, and Victor Yu. Novokshenov
Series: 
Mathematical Surveys and Monographs 128
Publication Date: 
2006
Format: 
Hardcover
Category: 
Monograph
  • Introduction. Painlevé transcendents as nonlinear special functions

Part 1. Riemannian-Hilbert problem, isomonodromy method and special functions

  • Systems of linear ordinary differential equations with rational coefficients. Elements of the general theory
  • Monodromy theory and special functions
  • Inverse monodromy problem and Riemann-Hilbert factorization
  • Isomonodromy deformations. The Painlevé equations
  • The isomonodromy method
  • Bäcklund transformations

Part 2. Asymptotics of the Painlevé II transcendent. A case study

  • Asymptotic solutions of the second Painlevé equation in the complex plane. Direct monodromy problem approach
  • Asymptotic solutions of the second Painlevé equation in the complex plane. Inverse monodromy problem approach
  • PII asymptotics on the canonical six-rays. The purely imaginary case
  • PII asymptotics on the canonical six-rays. Real-valued case
  • PII quasi-linear Stokes phenomenon

Part 3. Asymptotics of the third Painlevé transcendent

  • PIII equation, an overview
  • Sine-Gordon reduction of PIII
  • Canonical four-rays. Real-valued solutions of SG-PIII
  • Canonical four-rays. Singular solutions of the SG-PIII
  • Asymptotics in the complex plane of the SG-PIII transcendent
  • Proof of Theorem 3.4
  • The Birkhoff-Grothendieck theorem with a parameter
  • Bibliography
  • Subject index
Publish Book: 
Modify Date: 
Tuesday, October 24, 2006

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