Painleve Transcendents

Painleve Transcendents : The Riemann-Hilbert Approach

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At the turn of the twentieth century, the French mathematician Paul Painleve and his students classified second order nonlinear ordinary differential equations with the property that the location of possible branch points and essential singularities of their solutions does not depend on initial conditions. It turned out that there are only six such equations (up to natural equivalence), which later became known as Painleve I-VI. Although these equations were initially obtained answering a strictly mathematical question, they appeared later in an astonishing (and growing) range of applications, including, e.g., statistical physics, fluid mechanics, random matrices, and orthogonal polynomials. Actually, it is now becoming clear that the Painleve transcendents (i.e., the solutions of the Painleve equations) play the same role in nonlinear mathematical physics that the classical special functions, such as Airy and Bessel functions, play in linear physics.The explicit formulas relating the asymptotic behaviour of the classical special functions at different critical points play a crucial role in the applications of these functions. It is shown in this book that even though the six Painleve equations are nonlinear, it is still possible, using a new technique called the Riemann-Hilbert formalism, to obtain analogous explicit formulas for the Painleve transcendents. This striking fact, apparently unknown to Painleve and his contemporaries, is the key ingredient for the remarkable applicability of these 'nonlinear special functions'.The book describes in detail the Riemann-Hilbert method and emphasizes its close connection to classical monodromy theory of linear equations as well as to modern theory of integrable systems. In addition, the book contains an ample collection of material concerning the asymptotics of the Painleve functions and their various applications, which makes it a good reference source for everyone working in the theory and applications of Painleve equations and related areas.
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Product details

  • Hardback | 553 pages
  • 184.15 x 260.35 x 31.75mm | 1,143.05g
  • Providence, United States
  • English
  • 082183651X
  • 9780821836514

Table of contents

Introduction. Painleve 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 Painleve equations The isomonodromy method Backlund transformations Part 2. Asymptotics of the Painleve II transcendent. A case study: Asymptotic solutions of the second Painleve equation in the complex plane. Direct monodromy problem approach Asymptotic solutions of the second Painleve 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 Painleve 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 Appendix A. Proof of Theorem 3.4 Appendix B. The Birkhoff-Grothendieck theorem with a parameter Bibliography Subject index.
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