num_pages: 560
description: At the turn of the twentieth century, the French mathematician Paul PainlevÃ© 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 PainlevÃ© 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 PainlevÃ© transcendents (i.e., the solutions of the PainlevÃ© 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 PainlevÃ© equations are nonlinear, it is still possible, using a new technique called the Riemann-Hilbert formalism, to obtain analogous explicit formulas for the PainlevÃ© transcendents. This striking fact, apparently unknown to PainlevÃ© 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 PainlevÃ© functions and their various applications, which makes it a good reference source for everyone working in the theory and applications of Pa
language: English
title: Painleve Transcendents: The Riemann-hilbert Approach (Mathematical Surveys and Monographs)
titlecomplete: Painleve Transcendents: The Riemann-hilbert Approach (Mathematical Surveys and Monographs) (Mathematical Surveys and Monographs, 128)
publisher: American Mathematical Society