Painleve Differential Equations in the Complex Plane

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This book is the first comprehensive treatment of Painlevé differential equations in the complex plane. Starting with a rigorous presentation for the meromorphic nature of their solutions, the Nevanlinna theory will be applied to offer a detailed exposition of growth aspects and value distribution of Painlevé transcendents. The subsequent main part of the book is devoted to topics of classical background such as representations and expansions of solutions, solutions of special type like rational and special transcendental solutions, Bäcklund transformations and higher order analogues, treated separately for each of these six equations. The final chapter offers a short overview of applications of Painlevé equations, including an introduction to their discrete counterparts. Due to the present important role of Painlevé equations in physical applications, this monograph should be of interest to researchers in both mathematics and physics and to graduate students interested in mathematical physics and the theory of differential equations.

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Product Details

Meet the Author

Dr. Valerii Gromak works at the Belarussian State University in Minsk, Belarussia. Ilpo Laine is Professor at the Mathematics Department of the University of Joensuu, Finland. Shun Shimomura is Professor at the Mathematics Department of Keio University, Yokohama, Japan.

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Table of Contents

Introduction 1
1 Meromorphic nature of solutions 5
2 Growth of Painleve transcendents 32
3 Value distribution of Painleve transcendents 55
4 The first Painleve equation (P[subscript 1]) 78
5 The second Painleve equation (P[subscript 2]) 94
6 The fourth Painleve equation (P[subscript 4]) 121
7 The third Painleve equation (P[subscript 3]) 146
8 The fifth Painleve equation (P[subscript 5]) 178
9 The sixth Painleve equation (P[subscript 6]) 211
10 Applications of Painleve equations 247
App. A Local existence and uniqueness of solutions of complex differential equations 267
App. B Basic notations and facts in the Nevanlinna theory 274
Bibliography 283
Index 301
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