Numerical Solution of the Painlevé Equations

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Numerical Solution of the Painlevé Equations

 10 - 14 May 2010


  • Folkmar Bornemann, Technische Universität München
  • Peter Clarkson, University of Kent
  • Percy Deift, New York University
  • Alan Edelman, Massachusetts Institute of Technology
  • Alexander Its, Indiana University-Purdue University Indianapolis


The role that the classical special functions, such as the Airy, Bessel, Hermite, Legendre and hypergeometric functions, started to play in the 19th century, has now been greatly expanded by the Painlevé functions. Increasingly, as nonlinear science develops, people are finding that the solutions to an extraordinarily broad array of scientific problems, from neutron scattering theory, to partial differential equations, to transportation problems, to combinatorics, etc., can be expressed in terms of Painlevé transcendents. Much can be, and has been, proved regarding the algebraic and asymptotic properties of Painlevé transcendents. Here the role of integral representations and the classical steepest descent method in deriving precise asymptotics and connection formulae for the classical special functions is played, and expanded, by a Riemann-Hilbert representation of the Painlevé equations. The Riemann-Hilbert method is based on the observation that the Painlevé equations describe the isomonodromy deformations of certain systems of linear differential equations with rational coefficients, so solving a Painlevé equation is equivalent to solving an inverse monodromy problem. However on the other hand, very little is known, beyond some ad hoc calculations, about the numerical solution of the Painlevé equations. Writing useful software for nonlinear equations such as the Painlevé equations presents many challenges, conceptual, philosophical and technical. Without the help of linearity, it is not at all clear how to select a broad enough class of "representative problems''.


Alex Barnett, Dartmouth College - A New Integral Representation of Quasi-Periodic Fields and it Application to Scattering and Eigenvalue Problems
Folkmar Bornemann, Technische Universität München - Painlevé Representations in Random Matrix Theory: a Numerical Perspective   
Tom Claeys, Université de Lille 1 - Critical Asymptotics for Toeplitz Determinants   
Peter Clarkson, University of Kent - Numerics and Asymptotics for the Painlevé Equations  
Ovidiu Costin, Ohio State University - Adiabatic Invariants for Nonlinear ODEs and Applications to the Painlevé Equations  
Rodica Costin, Ohio State University - Representations of solutions of general nonlinear ODEs in singular regions
Percy Deift, New York University - The Painlevé Project – Goal and Methods  Percy Deift, New York University - Riemann-Hilbert Methods for Painlevé Equations     
Andrew Dienstfrey, National Institute of Standards and Technology - Numerical Solution of the Painlevé Equations  Boris Dubrovin, SISSA - Painlevé Transcendents and Hamiltonian PDEs
Alan Edelman, Massachusetts Institute of Technology - Numerical Software in Random Matrix Theory
Natasha Flyer, National Center for Atmospheric Research - Contour Integrations in the Complex Plane for the Study of Solutions to Initial Boundary Value Problems
Tamara Grava, SISSA - Painlevé Equations and Asymptotics
Nalini Joshi, University of Sydney - Geometric Asymptotics of the First Painlevé equation
Andrei Kapaev, St Petersburg State University of Service and Economy - On the Lax Pairs for the First Painlevé Equation
Christian Klein, Instituté de Mathématiques de Bourgogne - Dissipationless shocks and Painlevé equations     
Arno Kuijlaars, Katholieke Universiteit Leuven - The appearance of Painlevé II in a model of non intersecting paths
Davide Masoero, SISSA - Painleve I, Anharmonic Oscillators, WKB Analysis and Deformed TBA
Peter Miller, University of Michigan - Universal behaviour near separatrices in the semiclassical sine-Gordon equation  
Hartmut Monien, University of Bonn - Hankel Determinants of Zeta Functions
Victor Novokshenov, Russian Academy of Sciences - Padé Approximations of the Painlevé Transcendents    
Sheehan Olver, University of Oxford - Numerical Approximation of Riemann-Hilbert Problems: Painlevé II

Andre Weideman, University of Stellenbosch - Methods for Computing Complex Singularity Structure in Differential Equations