Painlevé Transcendents: Their Asymptotics and Physical Applications

ISBN-13: 9781489911605 / Angielski / Miękka / 2013 / 446 str.

Decio Levi; Pavel Winternitz

Opis

The NATO Advanced Research Workshop "Painleve Transcendents, their Asymp- totics and Physical Applications," held at the Alpine Inn in Sainte-Adele, near Montreal, September 2 -7, 1990, brought together a group of experts to discuss the topic and produce this volume. There were 41 participants from 14 countries and 27 lectures were presented, all included in this volume. The speakers presented reviews of topics to which they themselves have made important contributions and also re- sults of new original research. The result is a volume which, though multiauthored, has the character of a monograph on a single topic. This is the theory of nonlinear ordinary differential equations, the solutions of which have no movable singularities, other than poles, and the extension of this theory to partial differential equations. For short we shall call such systems "equations with the Painleve property." The search for such equations was a very topical mathematical problem in the 19th century. Early work concentrated on first order differential equations. One of Painleve's important contributions in this field was to develop simple methods applicable to higher order equations. In particular these methods made possible a complete analysis of the equation;; = f(y', y, x), where f is a rational function of y' and y, with coefficients that are analytic in x. The fundamental result due to Painleve (Acta Math.

Spis treści

I: Asymptotics of Painlevé Transcendents, Connection Formulas, New Mathematical Features.- Integral Equations and Connection Formulae for the Painlevé Equations.- Continuous and Discrete Painlevé Equations.- The Painlevé Transcendents as Nonlinear Special Functions.- Connection Results for the First Painlevé Equation.- Turning Points of Linear Systems and Double Asymptotics of the Painlevé Transcendents.- On the Linearization of the Third Painlevé Equation.- II: Painlevé Analysis and Integrability.- Differential Equations with Fixed Critical Points.- Unification of PDE and ODE Versions of Painlevé Analysis into a Single Invariant Version.- Painlevé Integrability: Theorems and Conjectures.- Riemann Double Waves, Darboux Method and the Painlevé Property.- Hirota Forms for the Six Painlevé Equations from Singularity Analysis.- Flexibility in Applying the Painlevé Test.- Insertion of the Darboux Transformation in the Invariant Painlevé Analysis of Nonlinear Partial Differential Equations.- Symmetries and Painlevé Property of Equations in the Burger’s Family.- The Singular Manifold Method.- III: Classification, Symmetries, Geometry, and Painlevé Equations.- Painlevé Balances and Dressing Transformations.- The Painlevé Classification of Partial Differential Equations.- Differential Invariants, Double Fibrations and Painlevé Equations.- The Painlevé Equations and the Dynkin Diagrams.- Singularities and Symmetries of Nonlinear Evolution Equations.- IV: Applications.- On Reductions of Self-Dual Yang-Mills Equations.- The Asymptotic Solution of the Stimulated Raman-Scattering Equation.- Symmetry Reduction for the Stimulated Raman Scattering Equations and the Asymptotics of Painlevé V via the Boutroux Transformation.- Stimulated Raman Scattering in the Transient Limit.- Spin Systems, Statistical Mechanics and Painlevé Functions.- Integrability of Chern-Simons-Higgs Vortex Equations and a Reduction of the Self-Dual Yang-Mills Equations to Three Dimensions.- Some Isomonodromy Problems in Hyperbolic Space.- Physical Applications of Painlevé Type Equations Quadratic in the Highest Derivatives.- Participants.- Author Index.