Softcover ISBN:  9781470475567 
Product Code:  SURV/128.S 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470413552 
Product Code:  SURV/128.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Softcover ISBN:  9781470475567 
eBook: ISBN:  9781470413552 
Product Code:  SURV/128.S.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 
Softcover ISBN:  9781470475567 
Product Code:  SURV/128.S 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470413552 
Product Code:  SURV/128.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Softcover ISBN:  9781470475567 
eBook ISBN:  9781470413552 
Product Code:  SURV/128.S.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 128; 2006; 553 ppMSC: Primary 34
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 RiemannHilbert 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 RiemannHilbert 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.
ReadershipGraduate students and research mathematicians interested in special functions, in particular, Painlevé transcendents.

Table of Contents

Chapters

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 RiemannHilbert 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 sixrays. The purely imaginary case

10. PII Asymptotics on the canonical sixrays. realvalued case

11. PII Quasilinear stokes phenomenon

12. PIII equation, an overview

13. SineGordon reduction of PIII

14. Canonical fourrays. Realvalued solutions of SGPIII

15. Canonical fourrays. Singular solutions of the SGPIII

16. Asymptotics in the complex plane of the SGPIII transcendent


Additional Material

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


RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Reviews
 Requests
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 RiemannHilbert 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 RiemannHilbert 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.

Chapters

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 RiemannHilbert 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 sixrays. The purely imaginary case

10. PII Asymptotics on the canonical sixrays. realvalued case

11. PII Quasilinear stokes phenomenon

12. PIII equation, an overview

13. SineGordon reduction of PIII

14. Canonical fourrays. Realvalued solutions of SGPIII

15. Canonical fourrays. Singular solutions of the SGPIII

16. Asymptotics in the complex plane of the SGPIII transcendent

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