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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.
List of contents
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.
Summary
Describes many physical phenomena, their analytic solutions, that in many cases are preferable to numerical computation, which may be long, costly and, worst, subject to numerical errors. In addition, the analytic approach provides a global knowledge of the solution, while the numerical approach is always local.
