Title: Painleve Transcendents: The Riemann-Hilbert Approach: 128.S (Mathematical Surveys and Monographs)
Publisher: American Mathematical Society
Publication Date: 2023
Language: English
ISBN 10: 1470475561
ISBN 13: 9781470475567
Binding: Soft cover
Condition: New

About this title

Synopsis

At the turn of the twentieth century, the French mathematician Paul Painlev� 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 Painlev� 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 Painlev� transcendents (i.e., the solutions of the Painlev� 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 Painlev� equations are nonlinear, it is still possible, using a new technique called the Riemann-Hilbert formalism, to obtain analogous explicit formulas for the Painlev� transcendents. This striking fact, apparently unknown to Painlev� 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 Painlev� functions and their various applications, which makes it a good reference source for everyone working in the theory and applications of Painlev� equations and related areas.

About the Author

Athanassios S. Fokas, Cambridge University, United Kingdom, Alexander R. Its, Indiana University-Purdue University Indianapolis, IN, Andrei A. Kapaev, and Victor Yu. Novokshenov, Russian Academy of Sciences, Ufa, Russia

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