Painlevé Transcendents: The Riemann-Hilbert Approach
About this Title
Athanassios S. Fokas, Cambridge University, Cambridge, United Kingdom, Alexander R. Its, Indiana University-Purdue University Indianapolis, Indianapolis, IN, Andrei A. Kapaev and Victor Yu. Novokshenov, Russian Academy of Sciences, Ufa, Russia
Publication: Mathematical Surveys and Monographs
Publication Year 2006: Volume 128
ISBNs: 978-0-8218-3651-4 (print); 978-1-4704-1355-2 (online)
MathSciNet review: MR2264522
MSC (2000): Primary 33E17; Secondary 30E25, 34M50, 34M55, 37K15, 37K20
At the turn of the twentieth century, the French mathematician Paul Painlevé 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 Painlevé 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 Painlevé transcendents (i.e., the solutions of the Painlevé 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 Painlevé equations are nonlinear, it is still possible, using a new technique called the Riemann-Hilbert formalism, to obtain analogous explicit formulas for the Painlevé transcendents. This striking fact, apparently unknown to Painlevé 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 Painlevé functions and their various applications, which makes it a good reference source for everyone working in the theory and applications of Painlevé equations and related areas.
Graduate students and research mathematicians interested in special functions, in particular, Painlevé transcendents.
Table of Contents
- 1. Systems of linear ordinary differential equations with rational coefficients. Elements of the general theory
- 2. Monodromy theory and special functions
- 3. Inverse monodromy problem and Riemann-Hilbert factorization
- 4. Isomonodromy deformations. The Painlevé equations
- 5. The isomonodromy method
- 6. Bäcklund transformations
- 7. Asymptotic solutions of the second Painlevé equation in the complex plane. Direct monodromy problem approach
- 8. Asymptotic solutions of the second Painlevé equation in the complex plane. Inverse monodromy problem approach
- 9. PII Asymptotics on the canonical six-rays. The purely imaginary case
- 10. PII Asymptotics on the canonical six-rays. real-valued case
- 11. PII Quasi-linear stokes phenomenon
- 12. PIII equation, an overview
- 13. Sine-Gordon reduction of PIII
- 14. Canonical four-rays. Real-valued solutions of SG-PIII
- 15. Canonical four-rays. Singular solutions of the SG-PIII
- 16. Asymptotics in the complex plane of the SG-PIII transcendent