Théories asymptotiques et équations de Painlevé

The history of Painlevé differential equations is more than a century long. They are second order nonlinear partial differential equations characterized by the behaviour of the poles of their solutions. Painlevé divided them into six families (P I - P VI) in his classification result. After a quiet period, they have recently become very popular in connection with their applications to important problems in theoretical physics. In the book, there are 14 articles connected with the meeting on Painlevé equations organized in Angers in 2004. A lot of the contributions are devoted to P VI: P. Boalch describes the group of its symmetries and some special solutions; D. Guzzetti discusses its elliptic representations; a long paper by M. Inaba, K. Iwasaki and M.-H. Saito describes its dynamics; and RS-transformations for construction of its algebraic solutions are discussed in the paper by A. V. Kitaev. The paper by P. A. Clarkson is devoted to P II, P III and P IV, while another paper by P. A. Clarkson, N. Joshi and M. Mazzocco discusses the (natural) Lax pair for the modified KdV hierarchy and its relation to isomonodromic problems for P II. The Painlevé property for the Hénon-Heiles Hamiltonians is the topic of the paper by R. Conte, M. Musette and C. Verhoeven. The papers by K. Kajiwara, T. Masuda, M. Nuomi, Y. Ohta and Y. Yamada, and by A. Ramani, B. Grammaticos and T. Tamizhmani are devoted to various aspects of discrete Painlevé equations. Integrability questions for P II are studied in a paper by J. J. Morales-Ruiz. A survey paper by J. Sauloy treats isomonodromy questions for complex, linear q-difference equations. Higher order Painlevé equations (and in particular Nuomi-Yamada systems) are discussed in the paper by Y. Takei. A study of Painlevé equations from the point of view of (infinite dimensional) Galois theory is written by H. Umemura. Asymptotic behaviour of solutions of linear, analytic q-difference equations is discussed in the paper by Ch. Zang.

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