The Painlevé Property: One Century Later

ISBN-13: 9780387988887 / Angielski / Twarda / 1999 / 810 str.

Robert Conte

Opis

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 http: //www . 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.

Spis treści

1 Singularities of Ordinary Linear Differential Equations and Integrability.- 1 Generalities.- 2 Structure of the Solutions of the Homogeneous Equation Around an Isolated Singular Point.- 3 Weakly Singular Equations (Fuchs).- 4 Thomé’s Equations.- 5 Global Considerations.- 6 References.- 2 Introduction to the Theory of Isomonodromic Deformations of Linear Ordinary Differential Equations with Rational Coefficients.- 1 Introduction.- 2 Isomonodromic Deformations of Linear ODEs with Puchsian Singularities.- 3 The Isomonodromic Deformation Problem for Painlevé VI.- 4 Isomonodromic Deformations of Linear ODEs with Thomé Singularities.- 5 An Isomonodromic Deformation Problem for Painlevé I.- 6 Conclusion.- 7 Appendix A: Matrix Versus Scalar Formalisms and Fuchs’s Theorem.- 8 Appendix B: Asymptotic Power Series.- 9 References.- 3 The Painlevé Approach to Nonlinear Ordinary Differential Equations.- 1 Introduction.- 2 The Meromorphy Assumption.- 3 The True Problems.- 4 The Classical Results (L. Fuchs, Poincaré, Painlevé).- 5 Construction of Necessary Conditions. The Theory.- 6 Construction of Necessary Conditions. The Painlevé Test.- 7 Sufficiency: Explicit Integration Methods.- 8 Conclusion.- 9 References.- 4 Asymptotic Studies of the Painlevé Equations.- 1 Introduction.- 2 Linear and Nonlinear Asymptotic Models.- 3 The First and Second Painlevé Equations.- 4 Global Extensions.- 5 Conclusion.- 6 References.- 5 2-D Quantum and Topological Gravities, Matrix Models, and Integrable Differential Systems.- A 2-D Quantum Gravity.- 1 Introduction.- 2 The One-Matrix Model: Large N Limit.- 3 The One-Matrix Model: Exact Solution.- 4 The Double-Scaling Limit.- 5 Multimatrix Models.- 6 Conclusion.- B 2-D Topological Gravity.- 7 Introduction.- 8 Computing the Kontsevich Integral.- 9 The Kontsevich Integral as T-Function of the KdV Hierarchy.- 10 Main Equivalence Theorem Between Topological and Quantum Gravities.- 11 Conclusion.- 12 References.- 6 Painlevé Transcendents in Two-Dimensional Topological Field Theory.- 1 Algebraic Properties of Correlators in 2-D Topological Field Theories. Moduli of a 2-D TFT and WDW Equations of Associativity.- 2 Equations of Associativity and Probenius Manifolds. Deformed Flat Connection and Its Monodromy at the Origin.- 3 Semisimplicity and Canonical Coordinates.- 4 Stokes Matrices and Classification of Semisimple Probenius Manifolds.- 5 Monodromy Group and Mirror Construction for Semisimple Frobenius Manifolds.- 6 References.- 7 Discrete Painlevé Equations.- 1 Integrable Discrete Systems.- 2 Similarity Reduction and Direct Linearization.- 3 The Painlevé Property for Discrete Systems.- 4 Properties of the Discrete Painlevé Equations.- 5 Monodromy Problems and q-Difference Equations.- 6 References.- 8 Painlevé Analysis for Nonlinear Partial Differential Equations 517.- 1 Introduction.- 2 Integrable Equations.- 3 Painlevé Analysis for PDEs.- 4 Partially Integrable and Nonintegrable Equations.- 5 References.- 9 On Painlevé and Darboux-Halphen-Type Equations.- 1 Introduction.- 2 Painlevé Equations and 1ST.- 3 Darboux-Halphen Systems and Their Linear Problems as Reductions of SDYM.- 4 The Monodromy Evolving System and the Solution of the Generalized DH System.- 5 Discussion.- 6 References.- 10 Symmetry Reduction and Exact Solutions of Nonlinear Partial Differential Equations.- 1 Introduction.- 2 Algorithm for Calculating the Symmetry Group of a Differential System.- 3 Examples of Symmetry Groups.- 4 Symmetry Reduction, Group Invariant Solutions, Partially Invariant Solutions.- 5 Classification of the Subalgebras of a Finite-Dimensional Lie Algebra.- 6 Direct Reductions and Conditional Symmetries.- 7 Conclusions.- 8 References.- 11 Painlevé Equations in Terms of Entire Functions.- 1 Introduction.- 2 Hirota’s Bilinear Method for Soliton Equations.- 3 Bilinear Forms and Similarity Reduction.- 4 Solutions in Terms of Entire Functions.- 5 Discrete Painlevé.- 6 References.- 12 Bäcklund Transformations of Painlevé Equations and Their Applications.- 1 Introduction.- 2 The Second Painlevé Equation.- 3 Rational Solutions of (Pin2) (0-Solutions).- 4 One-Parameter Families of Classical Solutions (1-Solutions).- 5 Algebraic Nonintegrability of (P2).- 6 Higher Analogue of (P2).- 7 The Fourth Painlevé Equation.- 8 Classical Solutions of (P4).- 9 Rational Solutions of (P4).- 10 The Third Painlevé Equation.- 11 Equation (P3) for ? = 0, ?? ? 0.- 12 Equation (P3) for ?? ? 0.- 13 Rational and Classical Solutions of (P3) for ?? ? 0.- 14 The Fifth Painlevé Equation.- 15 The Sixth Painlevé Equation.- 16 References.- 13 The Hamiltonians Associated to the Painlevé Equations.- 1 Introduction.- 2 Hamiltonians and Painlevé Analysis.- 3 The Space of Initial Conditions.- 4 The Irreducibility of PII.- 5 The T-Functions of the Second Painlevé System.- 6 The Painlevé System of Two Variables.- 7 References.- 14 “Completeness” of the Painlevé Test—General Considerations—Open Problems.- 1 Cultures in Mathematics.- 2 The Painlevé Test.- 3 The PolyPainlevé Test.- 4 Asymptotic Expansions.- 5 References.