The Painlevé Property

The Centre de recherches mathCmatiques (CRM) was created in 1968 by the Universite de Montreal to promote research in the mathematical sci ences. It is now a national institute that hosts several groups, holds special theme years, summer schools, workshops, postdoctoral program. The focus of its scientific activities ranges from pure to applied mathematics, and includes satistics, theoretical computer science, mathematical methods in biology and life sciences, and mathematical and theoretical physics. The CRM also promotes collaboration between mathematicians and industry. It is subsidized by the Natural Sciences and Engineering Research Council of Canada, the Fonds FCAR od the Province of Quebec, the Canadian Institute for Advanced Research and has private endowments. Current ac tivities, fellowships, and annual reports can be found on the CRM web page at . CRM. UMontreal. CAl. The CRM Series in Mathematical Physics will publish monographs, lec ture notes, and proceedings base on research pursued and events held at the Centre de recherches mathematiques. Yvan Saint-Aubin Montreal Preface The subject of this three-week school was the explicit integration, that is, analytical as opposed to numerical, of all kinds of nonlinear differential equations (ordinary differential, partial differential, finite difference). The result of such integration is ideally the "general solution," but there are numerous physical systems for which only a particular solution is accessible, for instance the solitary wave of the equation of Kuramoto and Sivashinsky in turbulence.

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The subject this volume is explicit integration, that is, the analytical as opposed to the numerical solution, of all kinds of nonlinear differential equations (ordinary differential, partial differential, finite difference). Such equations describe many physical phenomena, their analytic solutions (particular solutions, first integral, and so forth) are in many cases preferable to numerical computation, which may be long, costly and, worst, subject to numerical errors. In addition, the analytic approach can provide a global knowledge of the solution, while the numerical approach is always local. Explicit integration is based on the powerful methods based on an in-depth study of singularities, that were first used by Poincaré and subsequently developed by Painlevé in his famous Leçons de Stockholm of 1895. The recent interest in the subject and in the equations investigated by Painlevé dates back about thirty years ago, arising from three, apparently disjoint, fields: the Ising model of statistical physics and field theory, propagation of solitons, and dynamical systems. The chapters in this volume, based on courses given at Cargèse 1998, alternate mathematics and physics; they are intended to bring researchers entering the field to the level of present research.

Contenu
Singularities of Ordinary Linear Differential Equations.- Introduction to the Theory of Isomonodronic Deformations.- Painlevé Approach to Nonlinear Ordinary Differential Equations.- Asymptotic Studies of the Painlevé Equations.- 2D Quantum and Topological Gravities.- Painlevé Transcendents in Two Dimensional Topological Field.- Discrete Painlevé Equations.- Painlevé Analysis for Partial Differential Equations.- On Painlevé and Darboux Halpen Type Equations.- Symmetry Reduction and Exact Solutions.- Painlevé Equations in Terms of Entire Functions.- Backlund Transformations of Painlevé Equations.- The Hamiltonians Associated to Painleve Equations.- Completeness of the Painlevé Test.