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.

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.

Reviews
The book by Fokas et al. is a comprehensive, substantial, and impressive piece of work. Although much of the book is highly technical, the authors try to explain to the reader what they are trying to do. ... This book complements other monographs on the Painlevi equations. -- Journal of Approximation Theory The book is indispensable for both students and researchers working in the field. The authors include all necessary proofs of the results and the background material and, thus, the book is easy to read. -- Mathematical Reviews



Book Information
ISBN 9781470475567
Author Athanassios S. Fokas
Format Paperback
Page Count 553
Imprint American Mathematical Society
Publisher American Mathematical Society