Numerical Solution of the Painlevé Equations

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Numerical Solution of the Painlevé Equations

 10 - 14 May 2010
 14 India Street, Edinburgh

Scientific Organisers:

  • 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''.

Speakers:

  • Percy Deift, New York University - The Painlevé Project – Goal and Methods 

  • Andrew Dienstfrey, National Institute of Standards and Technology - Numerical Solution of the Painlevé Equations 

  • Ovidiu Costin, Ohio State University - Adiabatic Invariants for Nonlinear ODEs and Applications to the Painlevé Equations 

  • Boris Dubrovin, SISSA - Painlevé Transcendents and Hamiltonian PDEs

  • Sheehan Olver, University of Oxford - Numerical Approximation of Riemann-Hilbert Problems: Painlevé II

  • Tamara Grava, SISSA - Painlevé Equations and Asymptotics

  • Peter Clarkson, University of Kent - Numerics and Asymptotics for the Painlevé Equations 

  • Percy Deift, New York University - Riemann-Hilbert Methods for Painlevé Equations    

  • Folkmar Bornemann, Technische Universität München - Painlevé Representations in Random Matrix Theory: a Numerical Perspective  

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

  • Natasha Flyer, National Center for Atmospheric Research - Contour Integrations in the Complex Plane for the Study of Solutions to Initial Boundary Value Problems

  • Peter Miller, University of Michigan - Universal behaviour near separatrices in the semiclassical sine-Gordon equation 

  • Arno Kuijlaars, Katholieke Universiteit Leuven - The appearance of Painlevé II in a model of non intersecting paths

  • Rodica Costin, Ohio State University - Representations of solutions of general nonlinear ODEs in singular regions

  • Christian Klein, Instituté de Mathématiques de Bourgogne - Dissipationless shocks and Painlevé equations    

  • Alex Barnett, Dartmouth College - A New Integral Representation of Quasi-Periodic Fields and it Application to Scattering and Eigenvalue Problems

  • Hartmut Monien, University of Bonn - Hankel Determinants of Zeta Functions

  • Alan Edelman, Massachusetts Institute of Technology - Numerical Software in Random Matrix Theory

  • Tom Claeys, Université de Lille 1 - Critical Asymptotics for Toeplitz Determinants  

  • Victor Novokshenov, Russian Academy of Sciences - Padé Approximations of the Painlevé Transcendents    

  • Andrei Kapaev, St Petersburg State University of Service and Economy - On the Lax Pairs for the First Painlevé Equation

  • Nalini Joshi, University of Sydney - Geometric Asymptotics of the First Painlevé equation

  • Davide Masoero, SISSA - Painleve I, Anharmonic Oscillators, WKB Analysis and Deformed TBA

Participants

Name
Institution
Alex, BarnettDartmouth College
Lies, BoelenKatholieke Universiteit Leuven
Folkmar, BornemannTechnische Universität München
Anne, Boutet de MonvelUniversité Paris 7
Tom, ClaeysUniversité de Lille 1
Peter, ClarksonUniversity of Kent
Rodica, CostinOhio State University
Ovidiu, CostinOhio State University
Penny, DaviesUniversity of Strathclyde
Alfredo, DeañoUniversity of Kent
Percy, DeiftNew York University
Andrew, DienstfreyNational Institute of Standards and Technology
Boris, DubrovinSISSA
Dugald, DuncanHeriot-Watt University
Alan, EdelmanMassachusetts Institute of Technology
Natasha, FlyerNational Center for Atmospheric Research
Bengt, FornbergUniversity of Colorado
Tamara, GravaSISSA
Rod, HalburdUniversity College London
Alexander, ItsIndiana University-Purdue University Indianapolis
Nalini, JoshiUniversity of Sydney
Andrei, KapaevSt Petersburg State University of Service and Economy
Christian, KleinInstitut de Mathématiques de Bourgogne
Igor, KrasovskyBrunel University
Arno, KuijlaarsKatholieke Universiteit Leuven
Ben, LeimkuhlerUniversity of Edinburgh
Daniel, LozierNational Institute of Standards and Technology
Elizabeth, MansfieldUniversity of Kent
Davide, MasoeroSISSA
Peter, MillerUniversity of Michigan
Hartmut, MonienUniversity of Bonn
Lawrence, MulhollandNumerical Algorithms Group
Victor, NovokshenovRussian Academy of Sciences
Adri, Olde DaalhuisUniversity of Edinburgh
Sheehan, OlverUniversity of Oxford
Beatrice, PelloniHeriot-Watt University
Craig, TracyUC Davis
Andre, WeidemanUniversity of Stellenbosch