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) |
10.30 - 11.00 | Coffee/Tea |
11.00 - 11.45 | Dienstfrey, Andrew (National Institute of Standards and Technology) |
11.45 - 12.30 | Costin, Ovidiu (Ohio State University) |
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) |
17.15 - 18.30 | Informal wine reception in the Exhibition Room |
Tuesday 11 May
09.00 - 09.45 | Olver, Sheehan (University of Oxford) |
09.45 - 10.30 | Grava, Tamara (SISSA, Trieste) |
10.30 - 11.00 | Coffee/Tea |
11.00 - 11.45 | Clarkson, Peter (University of Kent) |
11.45 - 12.30 | Deift, Percy (New York University) |
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) |
Wednesday 12 May
09.00 - 09.45 | Weideman, Andre (University of Stellenbosch) |
09.45 - 10.30 | Flyer, Natasha (National Center for Atmospheric Research, Boulder) |
10.30 - 11.00 | Coffee/Tea |
11.00 - 11.45 | Edelman, Alan (Massachusetts Institute of Technology) |
11.45 - 12.30 | Claeys, Tom (Université de Lille 1) |
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) |
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) |
09.45 - 10.30 | Costin, Rodica (Ohio State University) |
10.30 - 11.00 | Coffee/Tea |
11.00 - 11.45 | Monien, Hartmut (University of Bonn) |
11.45 - 12.30 | Klein, Christian (Instituté de Mathématiques de Bourgogne) |
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) |
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) |
09.45 - 10.30 | Kapaev, Andrei (St Petersburg State University of Service and Economy) |
10.30 - 11.00 | Coffee/Tea |
11.00 - 11.45 | Joshi, Nalini (University of Sydney) |
11.45 - 12.30 | Masoero, Davide (SISSA, Trieste) |
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 | |
Bornemann, Folkmar | |
Painlevé representations in random matrix theory: a numerical perspective | |
View Abstract | |
Claeys, Tom | |
Critical asymptotics for Toeplitz determinants | |
View Abstract | |
Clarkson, Peter | |
Numerics and asymptotics for the Painlevé equations | |
View Abstract | |
Costin, Ovidiu | |
Adiabatic invariants for nonlinear ODEs and applications to the Painlevé equations | |
View Abstract | |
Costin, Rodica | |
Representations of solutions of general nonlinear ODEs in singular regions | |
View Abstract | |
Deift, Percy | |
2 talks: for further details see below in 'View Abstact' section | |
View Abstract | |
Dienstfrey, Andrew | |
Numerical solution of the Painlevé equations | |
View Abstract | |
Dubrovin, Boris | |
Painlevé transcendents and Hamiltonian PDEs | |
View Abstract | |
Edelman, Alan | |
Numerical software in random matrix theory | |
View Abstract | |
Flyer, Natasha | |
Contour integrations in the complex plane for the study of solutions to initial boundary value problems | |
View Abstract | |
Grava, Tamara | |
Painlevé equations and asymptotics | |
View Abstract | |
Joshi, Nalini | |
Geometric asymptotics of the first Painlevé equation | |
View Abstract | |
Kapaev, Andrei | |
On the Lax pairs for the first Painlevé equation. | |
View Abstract | |
Klein, Christian | |
Dissipationless shocks and Painlevé equations | |
View Abstract | |
Kuijlaars, Arno | |
The appearance of Painlevé II in a model of non intersecting paths | |
View Abstract | |
Masoero, Davide | |
Painlevé I, anharmonic oscillators, WKB analysis and Deformed TBA | |
View Abstract | |
Miller, Peter | |
Universal behaviour near separatrices in the semiclassical sine-Gordon equation | |
View Abstract | |
Monien, Hartmut | |
Hankel determinants of Zeta functions | |
View Abstract | |
Novokshenov, Victor | |
Padé approximations of the Painlevé transcendents | |
View Abstract | |
Olver, Sheehan | |
Numerical approximation of Riemann–Hilbert problems: Painlevé II | |
View Abstract | |
Weideman, Andre | |
Methods for computing complex singularity structure in differential equations | |
View Abstract |
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 | University of Kent |
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 | Heriot-Watt University |
Tracy, Craig | UC Davis |
Weideman, Andre | University of Stellenbosch |