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
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Many numerical problems in modern photonic and electromagnetic applications involve the interaction of linear waves with periodic, piecewise-homogeneous media. Boundary integral equations are a natural approach, with high efficiency and high-order or spectral convergence. The problem may be periodized by replacing the free-space Greens function with its quasi-periodic cousin; however, this method fails at parameter values corresponding to resonances of the empty unit cell. We fix this problem by combining the free space Greens function with auxiliary layer potentials on the lattice cell boundaries. We show numerical examples in two dimensions: the eigenvalue problem for Bloch waves in a photonic crystal, and the scattering of scalar waves from a grating, may both be solved robustly to 10 digits of accuracy with only a couple of hundred unknowns. The latter case uses layer potentials on unbounded curves, which may have utility in the integral-equation solution of Riemann-Hilbert problems. (Joint work with Leslie Greengard at NYU)
Bornemann, Folkmar
Painlevé representations in random matrix theory: a numerical perspective
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Suppose there is a general purpose software library for numerically evaluating Painlevé transcendents. Would it be of any use for the specific Painlevé representations of probability distributions encountered in Random Matrix Theory? Since those representations are typically based on separatrices of the Painlevé equations I will argue that for reasons of numerical stability this important additional information would have to be hard-coded into the numerical method; requiring an expert system for connection formulae and their accurate asymptotic expansions based on symbolic manipulation software. I will compare this strong requirement to the much simpler numerical evaluation of Fredholm determinant representations; a method which, however, has the structural drawback of not providing accurate tails.
Claeys, Tom
Critical asymptotics for Toeplitz determinants
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I will discuss asymptotic expansions for Toeplitz determinants corresponding to a family of symbols depending on a parameter t. For t positive, the symbols are regular and the determinants obey Szego's strong limit theorem. If t=0, the symbols have a Fisher-Hartwig singularity. Letting tto 0, we observe the emergence of a Fisher-Hartwig singularity and a transition between two different types of asymptotic behavior for Toeplitz determinants. This transition is described by a special Painlevé V transcendent. A particular case of our result describes a phase transition in the two-dimensional Ising model. The talk will be based on joint work with A. Its and I. Krasovsky.
Clarkson, Peter
Numerics and asymptotics for the Painlevé equations
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In this talk I shall discuss numerics, asymptotics and connection formulae for the Painlevé equations. In particular I shall exhibit some numerical evidence of new monotonically increasing and monotonically decreasing solutions of the second Painlevé equation, which exist on the whole real line and are generalizations of the Hastings-McLeod solution.
Costin, Ovidiu
Adiabatic invariants for nonlinear ODEs and applications to the Painlevé equations
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I will describe asymptotic methods stemming from transseries and Borel summability, applicable to relatively general nonlinear ODEs. They provide qualitative and quantitative control of the general solutions in wide regions of the complex domain and allow for efficient calculations of the solutions for large values of the independent variable. I will explain how the methods apply to Painlevé equations to numerically evaluate the solutions and to calculate connection constants.
Costin, Rodica
Representations of solutions of general nonlinear ODEs in singular regions
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I will discuss Borel summability methods for calculating decaying solutions of ODEs, and their extension to regions where they become large and develop singularities. Expansions in singular regions naturally extend to global representations of these solutions. An important application is in calculating Painlevé transcendents.
Deift, Percy
2 talks: for further details see below in 'View Abstact' section
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Talk 1: The Painlevé project - goal and methods Abstract to follow Talk 2: Riemann-Hilbert methods for Painlevé equations Abstract to follow
Dienstfrey, Andrew
Numerical solution of the Painlevé equations
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I will present an approach to the numerical solution of the Painlevé differential equations that closely follows the analysis of these equations as solutions to Riemann-Hilbert problems (RHP). Starting from the formulation of the RHP as a system of weighted singular integral equations, I will present a flexible, adaptive discretization of such equations and demonstrate their numerical solution. Furthermore, Riemann-Hilbert problems are amenable to diverse analytic solution treatments including: jump-matrix factorization, local solutions with contour deformation, and parametrix solutions. This flexibility is an asset that has been leveraged extensively to derive asymptotic analysis of the solutions to these equations. I will demonstrate that this freedom can provide benefit for numerical techniques as well. As an example I will present numerical results of the computation of the familiar Fredholm determinant of the sine-kernel, the solution of which is of interest in random matrix theory and furthermore is expressible with respect to a particular solution of Painlevé V.
Dubrovin, Boris
Painlevé transcendents and Hamiltonian PDEs
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To follow
Edelman, Alan
Numerical software in random matrix theory
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The creation of usable, algorithmically sound, numerical libraries is critical to the successful application of random matrix theory in far reaching fields. Many theoretical ideas need to become numerical experiments, experiments need to become reusable software, reusable software needs to be faster and more accurate. We will survey some developments and point out many opportunities.
Flyer, Natasha
Contour integrations in the complex plane for the study of solutions to initial boundary value problems
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A new method, combining complex analysis with numerics, is introduced for solving certain classes of partial differential equations (PDEs). The method yields novel integral representations, even for the solution of classical problems that would normally be solved via the Fourier or Laplace transforms. However, it has the advantage of being uniformly convergent in the limit as time goes to 0 for PDEs such as the heat equation, where the classical inverse Laplace integrals diverge. The numerical integration of the novel integral representations involves: 1) performing contour deformations in the complex plane such that the deformed integration paths are in regions where the integrands decay exponentially and 2) making a change of variables, which maps the contours from the complex plane to the real line in such a way that points concentrated in the regions where the integrands have greatest variability.
Grava, Tamara
Painlevé equations and asymptotics
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We study solutions of Hamiltonian PDEs at critical regimes. We show that these critical regimes are described by special solutions of Painlevé equations.
Joshi, Nalini
Geometric asymptotics of the first Painlevé equation
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I will report on my recent collaboration with Hans Duistermaat on the geometry of the space of initial values of the first Painlevé equation, which was first constructed by Okamoto. We show that highly accurate information about solutions can be found by utilizing the regularized and compactified space of initial values in Boutroux's coordinates. I will also describe numerical explorations based on this work obtained in collaboration with Holger Dullin.
Kapaev, Andrei
On the Lax pairs for the first Painlevé equation.
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Classical Painlevé equations admit various Lax pair representations. Here, we discuss the Lax pairs for the Painlevé first equation coming from the theory of orthogonal polynomials.
Klein, Christian
Dissipationless shocks and Painlevé equations
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The Cauchy problem for dissipationless equations as the Korteweg de Vries (KdV), Camassa-Holm (CH) and nonlinear Schrödinger (NLS) equation with small dispersion of order $epsilonll 1$, is characterized by the appearance of a zone of rapid modulated oscillations of wave-length of order $epsilon$. Near the gradient catastrophe of the dispersionless equation ($epsilon=0$), a multi-scales expansion gives an asymptotic solution in terms of a fourth order generalization of Painlevé I for KdV and CH, and of a Painlevé I transcendent for NLS. At the leading edge of the oscillatory zone for KdV and CH, a corresponding multi-scales expansion yields an asymptotic description of the oscillations where the envelope is given by a solution to the Painlevé II equation. We study the applicability of these approximations for several PDEs and random matrix models numerically.
Kuijlaars, Arno
The appearance of Painlevé II in a model of non intersecting paths
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We consider a model of n non-intersecting Brownian motions with two starting points and two ending points in the large n limit where the paths fill out two tangent ellipses. At the point of tangency a new critical behaviour occurs that is described by a 4 x 4 model Riemann-Hilbert problem. The solution of the Riemann-Hilbert problem is governed by the Painlevé II equation. (This is work in progress, jointly with Steven Delvaux and Lun Zhang).
Masoero, Davide
Painlevé I, anharmonic oscillators, WKB analysis and Deformed TBA
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Poles of solutions to the Painlevé-I equation are intimately related to the theory of the cubic anharmonic oscillator. In particular, poles of integrale tritronquee are in 1-1 correspondence with cubic oscillators that admit the simultaneous solutions of two quantization conditions. We analyze this pair of quantization conditions by means of a suitable version of the complex WKB method. Eventually We introduce a new tool, the Deformed TBA (Deformed Thermodynamic Bethe Ansatz), to analyze the monodromy problem of the cubic oscillator.
Miller, Peter
Universal behaviour near separatrices in the semiclassical sine-Gordon equation
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The semiclassically scaled sine-Gordon equation is considered with (relatively) slowly varying initial data. Considered as functions of x, the initial conditions parametrize a curve in the phase portrait of the simple pendulum. We study the solution of the initial-value problem near points x where the initial data crosses the pendulum separatrix, and show that the asymptotic behavior is independent of details of the initial data. This is joint work with Robert Buckingham.
Monien, Hartmut
Hankel determinants of Zeta functions
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We derive a general expression for the Hankel determinants of a Dirichlet series F(s) and determine the asymptotic behavior for the special case of the Riemann zeta function. In this case the Hankel determinant is a discrete analogue of the Selberg integral and can be viewed as a matrix integral with discrete measure.
Novokshenov, Victor
Padé approximations of the Painlevé transcendents
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All solutions to the Painlevé equations are meromorphic functions. However, no reliable algorithm is known to approximate Painlevé transcendents in the complex plane. It is natural to choose the algorithm as the (main diagonal) Padé approximation, since it is convergent to the meromorphic function in any compact domain. A version of the Fair-Luke algorithm has been applied to find the Padé approximate solutions to the Painlevé I and II equations. The distributions of poles for the famous Hastings-McLeod and Ablowitz-Segur solutions to PII equation were found. The Boutroux tritronqueé solution to PI equation was shown to have poles only in the critical sector of the complex plane. The algorithm allows to check further conjections on other analytic properties of the Painlevé transcendents in the complex plane.
Olver, Sheehan
Numerical approximation of Riemann–Hilbert problems: Painlevé II
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We describe a new spectral method for solving matrix-valued Riemann–Hilbert problems numerically. We demonstrate the effectiveness of this approach by computing solutions to the homogeneous Painlevé II equation. This can be used to relate initial conditions with asymptotic behaviour. Comparison with known values for the Hastings–McLeod solution shows that the method converges spectrally fast. By applying the approach to the deformed Riemann–Hilbert problem from nonlinear steepest descent, we can compute solutions for large values as well as small.
Weideman, Andre
Methods for computing complex singularity structure in differential equations
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We discuss numerical methods for computing complex singularity structure in ODEs and PDEs. These methods are based on Padé approximation and the epsilon algorithm. Examples include blow-up in a nonlinear heat equation and shock formation in the Burgers' equation. In keeping with the theme of the workshop we plan to present results on the Painlevé equation as well.

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