# 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

*- 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, Barnett | Dartmouth College |

Lies, Boelen | Katholieke Universiteit Leuven |

Folkmar, Bornemann | Technische Universität München |

Anne, Boutet de Monvel | Université Paris 7 |

Tom, Claeys | Université de Lille 1 |

Peter, Clarkson | University of Kent |

Rodica, Costin | Ohio State University |

Ovidiu, Costin | Ohio State University |

Penny, Davies | University of Strathclyde |

Alfredo, Deaño | University of Kent |

Percy, Deift | New York University |

Andrew, Dienstfrey | National Institute of Standards and Technology |

Boris, Dubrovin | SISSA |

Dugald, Duncan | Heriot-Watt University |

Alan, Edelman | Massachusetts Institute of Technology |

Natasha, Flyer | National Center for Atmospheric Research |

Bengt, Fornberg | University of Colorado |

Tamara, Grava | SISSA |

Rod, Halburd | University College London |

Alexander, Its | Indiana University-Purdue University Indianapolis |

Nalini, Joshi | University of Sydney |

Andrei, Kapaev | St Petersburg State University of Service and Economy |

Christian, Klein | Institut de Mathématiques de Bourgogne |

Igor, Krasovsky | Brunel University |

Arno, Kuijlaars | Katholieke Universiteit Leuven |

Ben, Leimkuhler | University of Edinburgh |

Daniel, Lozier | National Institute of Standards and Technology |

Elizabeth, Mansfield | University of Kent |

Davide, Masoero | SISSA |

Peter, Miller | University of Michigan |

Hartmut, Monien | University of Bonn |

Lawrence, Mulholland | Numerical Algorithms Group |

Victor, Novokshenov | Russian Academy of Sciences |

Adri, Olde Daalhuis | University of Edinburgh |

Sheehan, Olver | University of Oxford |

Beatrice, Pelloni | Heriot-Watt University |

Craig, Tracy | UC Davis |

Andre, Weideman | University of Stellenbosch |