The Painvele Property - One Century Later

The subject of 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, and 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 rounding errors. In addition, the analytic approach can provide a global knowledge of the solution, while the numerical approach is always local. Explicit integration uses 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 to about thirty years ago, arising from three, apparently disjoint, fields: the Ising model and other models of statistical physics and field theory, propagation of solitons, and dynamical systems. The chapters in this volume, based on courses given at Cargèse, alternate mathematics and physics. they are intended to bring researchers entering the field up to date with current research.