**Mathematical Surveys and Monographs**

Volume: 128;
2006;
553 pp;
Hardcover

MSC: Primary 34;

Print ISBN: 978-0-8218-3651-4

Product Code: SURV/128

List Price: $112.00

AMS Member Price: $89.60

MAA Member Price: $100.80

**Electronic ISBN: 978-1-4704-1355-2
Product Code: SURV/128.E**

List Price: $112.00

AMS Member Price: $89.60

MAA Member Price: $100.80

#### Supplemental Materials

# Painlevé Transcendents: The Riemann-Hilbert Approach

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*Athanassios S. Fokas; Alexander R. Its; Andrei A. Kapaev; Victor Yu. Novokshenov*

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.

#### Readership

Graduate students and research mathematicians interested in special functions, in particular, Painlevé transcendents.

#### Reviews & Endorsements

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

#### Table of Contents

# Table of Contents

## Painleve Transcendents: The Riemann-Hilbert Approach

- Contents vii8 free
- Preface xi12 free
- Introduction. Painlevé Transcendents as Nonlinear Special Functions 114 free
- Part 1. Riemann-Hilbert Problem, Isomonodromy Method and Special Functions 3750 free
- Chapter 1. Systems of Linear Ordinary Differential Equations with Rational Coefficients. Elements of the General Theory 3952
- Chapter 2. Monodromy Theory and Special Functions 6578
- Chapter 3. Inverse Monodromy Problem and Riemann-Hilbert Factorization 97110
- Chapter 4. Isomonodromy Deformations. The Painlevé Equations 133146
- Chapter 5. The Isomonodromy Method 161174
- Chapter 6. Bäcklund Transformations 219232

- Part 2. Asymptotics of the Painlevé II Transcendent. A Case Study 231244
- Chapter 7. Asymptotic Solutions of the Second Painlevé Equation in the Complex Plane. Direct Monodromy Problem Approach 233246
- 0. Introduction 233246
- 1. Preliminary remarks. The Boutroux ansatz 234247
- 2. Direct monodromy problem. Formulation of the main theorem 237250
- 3. WKB-analysis for the Ψ-function. 240253
- 4. Local solutions near turning points 245258
- 5. Approximation of the monodromy data 247260
- 6. Uniformization by theta-functions. Justification of the Boutroux ansatz 252265

- Chapter 8. Asymptotic Solutions of the Second Painlevé Equation in the Complex Plane. Inverse Monodromy Problem Approach 261274
- 0. Preliminary remarks. The Deift-Zhou method 261274
- 1. RH parametrization of PII in the complex plane 265278
- 2. The transformation of the RH problem 268281
- 3. Construction of the function g(z) 276289
- 4. The model Baker-Akhiezer RH problem 278291
- 5. Local RH problems near the branch points 283296
- 6. Asymptotic solution of the main RH problem 290303
- 7. Asymptotics of the Painlevé function 295308

- Chapter 9. PII Asymptotics on the Canonical Six-rays. The Purely Imaginary Case 297310
- 0. Introduction 297310
- 1. Formulation of results. Main theorems and discussion 298311
- 2. The direct monodromy problem approach. The proof of Theorem 9.3 303316
- 3. The direct monodromy problem approach. The proof of Theorem 9.4 313326
- 4. The inverse monodromy problem approach. The proof of Theorem 9.1 318331
- 5. The inverse monodromy problem approach. The proof of Theorem 9.2 331344

- Chapter 10. PII Asymptotics on the Canonical Six-rays. Real-valued Case 349362
- Chapter 11. PII Quasi-linear Stokes Phenomenon 373386
- 0. Introduction 373386
- 1. Linear Stokes phenomenon revisited 374387
- 2. General remarks on the Stokes phenomenon for PII 379392
- 3. The RH problem for PII 381394
- 4. Special points of the monodromy surface 387400
- 5. Nonspecial points of the monodromy surface 394407
- 6. Asymptotic solution for s[sub(2)] = 0 412425
- 7. The Hastings-McLeod solution 428441

- Part 3. Asymptotics of the Third Painlevé Transcendent 433446
- Chapter 12. PIII Equation, an Overview 435448
- Chapter 13. Sine-Gordon Reduction of PIII 443456
- Chapter 14. Canonical Four-rays. Real-valued Solutions of SG-PIII 459472
- Chapter 15. Canonical Four-rays. Singular Solutions of the SG-PIII 479492
- Chapter 16. Asymptotics in the Complex Plane of the SG-PIII Transcendent 495508

- Appendix A. Proof of Theorem 3.4 511524
- Appendix B. The Birkhoff-Grothendieck Theorem with a Parameter 531544
- Bibliography 539552
- Subject Index 551564