Entrance hall of the ICMS

Numerical solution of the Painlevé equations

May 10, 2010 - May 14, 2010

14 India Street, Edinburgh, EH3 6EZ

Organisers

Name Institution
Bornemann, Folkmar Technische Universität München
Clarkson, Peter University of Kent
Deift, Percy New York University
Edelman, Alan Massachusetts Institute of Technology
Its, Alexander 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''.

The Painlevé equations, which were discovered about a hundred years ago by Painlevé, Gambier and their colleagues, are special amongst nonlinear ordinary differential equations in that they are "integrable" due to their representation as Riemann-Hilbert problems. Further they can be thought of as nonlinear analogues of the classical special functions and have a plethora of remarkable properties. An important question is how these properties of the Painlevé equations, in particular their representation as Riemann-Hilbert problems, can be used in the development of software for the numerical solution of the equations.

The final report for this workshop is available here.

Arrangements

Participation
Participation is by invitation only.

UK Visas
If you are travelling from overseas you may require an entry visa. A European visa does not guarantee entry to the UK. Please use this link to the UK Visas site to find out if you need a visa and, if so, how to apply for one. If you do require a visa, ICMS can provide a signed invitation letter.

Travel Insurance
Due to recent difficulties with the volcanic ash cloud from Iceland disrupting air travel over the United Kingdom, please be aware that it is wise to have travel insurance in place for your trip. 

Venue
The workshop will take place at 14 India Street, Edinburgh, EH3 6EZ. This house is the birthplace of James Clerk Maxwell and is situated in the historic New Town of Edinburgh, near the city centre.

The ICMS travel pages contain advice on how to travel to Edinburgh. To find the location of the workshop venue follow this link to 14 India Street.

If you have been invited to give a presentation, the seminar room at 14 India Street has whiteboards, 2 overhead projectors, a data projector and laptop. We ask that you use our laptop for your presentation which you may bring on a memory stick or CD. Alternatively, a pdf or PowerPoint file may be emailed to Audrey Brown by Friday 7 May to be loaded onto the laptop in advance of the workshop.

Wireless access is available throughout the building. There are also 2 public PCs which may be used at any time for internet access and to check email.

Accommodation
ICMS will arrange en-suite rooms in hotels/guest houses nearby for those who request this. Accommodation is typically about 10 to 20 minutes walk from 14 India Street. Participants making their own arrangements may claim back the cost, with original receipts, up to a maximum of £70.00 per night bed and breakfast for a maximum of five or six nights. A list of Edinburgh accommodation of various sorts and prices is available here . Sections 1-3 are particularly relevant.

Meals and Refreshments
Morning and afternoon refreshments will be provided on each day of the workshop.

On Monday 10 May, the first day of the workshop, a light buffet lunch will be provided free of charge to participants in the Exhibition Room at 14 India Street. Thereafter, participants are free to explore the many cafés, sandwich shops, restaurants and bars nearby.  There will be an informal wine reception on Monday 10 May after the talks.   On Wednesday 12 May at 17.30, Cambridge University Press will host an informal wine reception.  This will be followed by an evening meal at a local Sicilian restaurant, Ristorante Librizzi.  The Workshop Dinner will take place on the evening of Thursday 13 May in The Magnum Restaurant.   The workshop grant will cover the cost of both of these dinners.

Registration
Registration will take place from 09.00 - 09.45 on Monday 10 May in the Exhibition Room at 14 India Street.

Financial Arrangements
Unless otherwise specified in your invitation letter, the workshop grant will cover the cost of your bed and breakfast accommodation, tea/coffee Monday to Friday, lunch on Monday only, an informal wine reception on Monday evening, an evening meal on Wednesday and the Workshop Dinner on Thursday.

If we have agreed to pay some of your travel costs, you will be informed by email. Reimbursement will take place after the workshop and will involve payment directly into your bank account. At Registration you will be given an expenses claim form and this should be submitted to ICMS, with original receipts. It would be helpful if you could bring your bank details to the workshop. In addition to the bank account number, participants from the USA and Canada will require their bank’s routing number, those from the UK will be asked for the bank sort code, and those from Europe and the rest of the world, their IBAN and SWIFT/BIC code. We cannot reimburse any item without a signed claim form and original receipts.

Unless otherwise specified by email, a registration fee of 30.00 GBP is payable by all participants. This may be paid at Registration, either in cash, by sterling cheque or by completing a form with your credit/debit card details.  The credit and debit card transactions will be made the following day and a receipt provided. 

 

Programme

Please continue to check the Programme as any alterations will be notified here.

Discussion Sessions

Monday The Painlevé project - goal and methods Deift
Tuesday Asymptotics methods for the Painlevé equations Its
Wednesday Random Matrices & Painlevé equations I Bornemann
Thursday Random Matrices & Painlevé equations II Edelman
Friday Open Problems Clarkson



Session Chairs

  Morning Afternoon
Monday Clarkson Deift
Tuesday Pelloni Its
Wednesday Tracy Bornemann
Thursday Lozier Edelman
Friday Halburd Clarkson
 

 

Monday 10 May

09.00 - 09.45

Registration and welcome

09.45 - 10.30

Deift, Percy (New York University)
The Painlevé project – goal and methods  pdf of presentation 

10.30 - 11.00

Coffee/Tea

11.00 - 11.45

Dienstfrey, Andrew (National Institute of Standards and Technology)
Numerical solution of the Painlevé equations 
pdf of presentation

11.45 - 12.30

Costin, Ovidiu (Ohio State University)
Adiabatic invariants for nonlinear ODEs and applications to the Painlevé equations  pdf of presentation

12.30 - 14.00

Lunch provided in the Exhibition Room

14.00 - 16.00

Discussion

16.00 - 16.30

Coffee/Tea

16.30 - 17.15

Dubrovin, Boris (SISSA, Trieste)
Painlevé transcendents and Hamiltonian PDEs

17.15 - 18.30

Informal wine reception in the Exhibition Room


Tuesday 11 May

09.00 - 09.45

Olver, Sheehan (University of Oxford)
Numerical approximation of Riemann-Hilbert problems: Painlevé II

09.45 - 10.30

Grava, Tamara (SISSA, Trieste)
Painlevé equations and asymptotics

10.30 - 11.00

Coffee/Tea

11.00 - 11.45

Clarkson, Peter (University of Kent)
Numerics and asymptotics for the Painlevé equations  pdf of presentation

11.45 - 12.30

Deift, Percy (New York University)
Riemann-Hilbert methods for Painlevé equations   pdf of presentation

12.30 - 14.00

Lunch break

14.00 - 16.00

Discussion  pdf of discussion presentation - Craig Tracy

16.00 - 16.30

Coffee/Tea

16.30 - 17.15

Bornemann, Folkmar (Technische Universität München)
Painlevé representations in random matrix theory: a numerical perspective  pdf of presentation


Wednesday 12 May

09.00 - 09.45

Weideman, Andre (University of Stellenbosch)
Methods for computing complex singularity structure in differential equations  pdf of presentation

09.45 - 10.30

Flyer, Natasha (National Center for Atmospheric Research, Boulder)
Contour integrations in the complex plane for the study of solutions to initial boundary value problems

10.30 - 11.00

Coffee/Tea

11.00 - 11.45

Edelman, Alan (Massachusetts Institute of Technology)
Numerical software in random matrix theory

11.45 - 12.30

Claeys, Tom (Université de Lille 1)
Critical asymptotics for Toeplitz determinants  pdf of presentation

12.30 - 14.00

Lunch break

14.00 - 16.00

Discussion pdf of discussion (by Folkmar Bornemann)

16.00 - 16.30

Coffee/Tea

16.30 - 17.15

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

 17.30 - 18.45

The Digital Library of Mathematical Functions project and informal wine reception hosted by Cambridge University Press.   

 19.00

Informal evening meal - at Ristorante Librizzi (a Sicilian restaurant), 69 North Castle Street.  (This supper will be provided free of charge to workshop participants).    


Thursday 13 May

09.00 - 09.45

Miller, Peter (University of Michigan)
Universal behaviour near separatrices in the semiclassical sine-Gordon equation  pdf of presentation and link to video

09.45 - 10.30

Costin, Rodica (Ohio State University)
Representations of solutions of general nonlinear ODEs in singular regions

10.30 - 11.00

Coffee/Tea

11.00 - 11.45

Monien, Hartmut (University of Bonn)
Hankel determinants of Zeta functions

11.45 - 12.30

Klein, Christian (Instituté de Mathématiques de Bourgogne)
Dissipationless shocks and Painlevé equations   pdf of presentation

12.30 - 14.00

Lunch break

14.00 - 16.00

Discussion

16.00 - 16.30

Coffee/Tea

16.30 - 17.15

Barnett, Alex (Dartmouth College)
A new integral representation of quasi-periodic fields and it application to scattering and eigenvalue problems

19.00

Workshop Dinner at The Magnum Restaurant, 1 Albany Street, Edinburgh


Friday 14 May

09.00 - 09.45

Novokshenov, Victor (Russian Academy of Sciences)
Padé approximations of the Painlevé transcendents    pdf of presentation

09.45 - 10.30

Kapaev, Andrei (St Petersburg State University of Service and Economy)
On the Lax pairs for the first Painlevé equation

10.30 - 11.00

Coffee/Tea

11.00 - 11.45

Joshi, Nalini (University of Sydney)
Geometric asymptotics of the first Painlevé equation

11.45 - 12.30

Masoero, Davide (SISSA, Trieste)
Painleve I, anharmonic oscillators, WKB analysis and Deformed TBA

2.30 - 14.00

Lunch break

14.00 - 16.00

Discussion

16.00 - 16.30

Coffee/Tea - End of workshop


Presentations:

Presentation Details
Barnett, Alex
A new integral representation of quasi-periodic fields and it application to scattering and eigenvalue problems
View Abstract Down
Bornemann, Folkmar
Painlevé representations in random matrix theory: a numerical perspective
View Abstract Down
Claeys, Tom
Critical asymptotics for Toeplitz determinants
View Abstract Down
Clarkson, Peter
Numerics and asymptotics for the Painlevé equations
View Abstract Down
Costin, Ovidiu
Adiabatic invariants for nonlinear ODEs and applications to the Painlevé equations
View Abstract Down
Costin, Rodica
Representations of solutions of general nonlinear ODEs in singular regions
View Abstract Down
Deift, Percy
2 talks: for further details see below in 'View Abstact' section
View Abstract Down
Dienstfrey, Andrew
Numerical solution of the Painlevé equations
View Abstract Down
Dubrovin, Boris
Painlevé transcendents and Hamiltonian PDEs
View Abstract Down
Edelman, Alan
Numerical software in random matrix theory
View Abstract Down
Flyer, Natasha
Contour integrations in the complex plane for the study of solutions to initial boundary value problems
View Abstract Down
Grava, Tamara
Painlevé equations and asymptotics
View Abstract Down
Joshi, Nalini
Geometric asymptotics of the first Painlevé equation
View Abstract Down
Kapaev, Andrei
On the Lax pairs for the first Painlevé equation.
View Abstract Down
Klein, Christian
Dissipationless shocks and Painlevé equations
View Abstract Down
Kuijlaars, Arno
The appearance of Painlevé II in a model of non intersecting paths
View Abstract Down
Masoero, Davide
Painlevé I, anharmonic oscillators, WKB analysis and Deformed TBA
View Abstract Down
Miller, Peter
Universal behaviour near separatrices in the semiclassical sine-Gordon equation
View Abstract Down
Monien, Hartmut
Hankel determinants of Zeta functions
View Abstract Down
Novokshenov, Victor
Padé approximations of the Painlevé transcendents
View Abstract Down
Olver, Sheehan
Numerical approximation of Riemann–Hilbert problems: Painlevé II
View Abstract Down
Weideman, Andre
Methods for computing complex singularity structure in differential equations
View Abstract Down

Participants

Name Institution
Barnett, Alex Dartmouth College
Boelen, Lies Katholieke Universiteit Leuven
Bornemann, Folkmar Technische Universität München
Boutet de Monvel, Anne Université Paris 7
Claeys, Tom Université de Lille 1
Clarkson, Peter University of Kent
Costin, Ovidiu Ohio State University
Costin, Rodica Ohio State University
Davies, Penny University of Strathclyde
Deaño, Alfredo Universidad Carlos III de Madrid
Deift, Percy New York University
Dienstfrey, Andrew National Institute of Standards and Technology
Dubrovin, Boris SISSA
Duncan, Dugald Heriot-Watt University
Edelman, Alan Massachusetts Institute of Technology
Flyer, Natasha National Center for Atmospheric Research
Fornberg, Bengt University of Colorado
Grava, Tamara SISSA
Halburd, Rod University College London
Its, Alexander Indiana University-Purdue University Indianapolis
Joshi, Nalini University of Sydney
Kapaev, Andrei St Petersburg State University of Service and Economy
Klein, Christian Institut de Mathématiques de Bourgogne
Krasovsky, Igor Brunel University
Kuijlaars, Arno Katholieke Universiteit Leuven
Leimkuhler, Ben University of Edinburgh
Lozier, Daniel National Institute of Standards and Technology
Mansfield, Elizabeth University of Kent
Masoero, Davide SISSA
Miller, Peter University of Michigan
Monien, Hartmut University of Bonn
Mulholland, Lawrence Numerical Algorithms Group
Novokshenov, Victor Russian Academy of Sciences
Olde Daalhuis, Adri University of Edinburgh
Olver, Sheehan University of Oxford
Pelloni, Beatrice University of Reading
Tracy, Craig UC Davis
Weideman, Andre University of Stellenbosch