Discrete integrable systems

Programme theme

The theory of (ordinary and partial) differential equations is well-established and to some extent standardised. By contrast, the theory of difference equations, while more fundamental, has until recently been in its infancy, in spite of a major effort at the beginning of the 20th Century by N"orlund and the school of G.D. Birkhoff to establish the linear theory. Discrete systems can appear in two main guises: in the first case the independent variable is discrete, taking values on a lattice (e.g. finite-difference equations, such as recurrence relations and dynamical mappings), in the second case the independent variable is continuous (e.g. analytic difference equations and even functional equations).

Very recently, however, mainly through advances in the theory of exactly integrable discrete systems and the theory of (linear and nonlinear) special functions, the study of difference equations has undergone a true revolution. For the first time good and interesting examples of nonlinear difference equations admitting exact, albeit highly nontrivial, solutions were found and this has led to the formulation of novel approaches to the classification and treatment of such equations. Thus, an area has developed where several branches of mathematics and physics, that are usually distinct, come together: complex analysis, algebraic geometry, representation theory, Galois theory, spectral analysis and the theory of special functions, graph theory, and difference geometry.

Looking at recent developments in these various fields one observes that many of the areas that are clearly on the verge of major breakthroughs have an intimate connection with discrete integrable systems. Such fields comprise: the algebraic geometry of rational surfaces and birational maps, difference Galois theory and isomonodromic deformation theory, Diophantine problems in number theory and p-adic analysis, quantum and non-commutative discrete integrable systems, representation theory of quantum algebras and associated cluster algebras as well as special functions, difference geometry and symmetries and conservation laws of discrete systems. Bringing together experts from these fields for a substantial period of time the programme aims at bringing about important cross-fertilisation of ideas and methodologies, and thereby substantial new advances in the theory of difference equations.

The Institute kindly requests that any papers published as a result of this programme’s activities are credited as such. Please acknowledge the support of the Institute in your paper using the following text:

The author(s) would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Discrete integrable systems, where work on this paper was undertaken. This work was supported by EPSRC grant EP/F005431/1.

Click here to download the programme's final scientific report

Preprints

Title	Year	Programme	Publication Date
On a nonlinear integrable difference equation on the square 3D-inconsistent Authors: Ravil Yamilov, Decio Levi Download PDF	2008	DIS	21 October 2016
The generalised symmetry method for discrete equations Authors: Ravil Yamilov, Decio Levi Download PDF	2008	DIS	21 October 2016
Soliton solutions for ABS lattice equations: I Cauchy matrix approach Authors: James Atkinson, Frank Nijhoff, Jarmo Hietarinta Download PDF	2008	DIS	21 October 2016
Tzitzeica solitons vs. relativistic Calogero-Moser 3-body clusters Authors: Simon Ruijsenaars, Jonathan Nimmo Download PDF	2008	DIS	21 October 2016
Symmetries and integrability of discrete equations defined on a black-white lattice Authors: Vassilios Papageorgiou, Pavlos Xenitidis Download PDF	2008	DIS	21 October 2016
Singular finite-gap operators and indefinite metric. I. Authors: Sergey Novikov, P Grinevich Download PDF	2008	DIS	21 October 2016
Lagrangian multiforms and multidimensional consistency Authors: Frank Nijhoff, Sarah Lobb Download PDF	2008	DIS	21 October 2016
Integrability, analyticity, isochrony, equilibria, small oscillations and Diophantine relations Authors: Francesco Calogero Download PDF	2008	DIS	21 October 2016
Three-dimensional discrete systems of Hirota-Kimura type and deformed Lie-Poisson algebras Authors: Matteo Petrera, ANW Hone Download PDF	2008	DIS	21 October 2016
Linear quadrilateral lattice equations and multidimensional consistency Authors: James Atkinson Download PDF	2008	DIS	21 October 2016
Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa$-$Holm type equation Authors: H Lundmark, J Szmigielski, ANW Hone Download PDF	2008	DIS	21 October 2016
On the Neimark$-$Sacker bifurcation in a discrete predator-prey system Authors: MV Irle, GW Thurura, ANW Hone Download PDF	2008	DIS	21 October 2016
Note on the Poisson structure of the damped oscillator Authors: M Senthilvelan, ANW Hone Download PDF	2008	DIS	21 October 2016
On the non-integrability of the Popowicz peakon system Authors: MV Irle, ANW Hone Download PDF	2008	DIS	21 October 2016
On non-standard numerical integration methods for biological oscillators Authors: ANW Hone Download PDF	2008	DIS	21 October 2016
A self-adaptive mesh method for the Camassa$-$Holm equation Authors: Yasuhiro Ohta, K Maruno, BF Feng Download PDF	2008	DIS	21 October 2016
Holomorphic curves with shift-invariant hyperplane preimages Authors: R Korhonen, K Tohge, Rod Halburd Download PDF	2008	DIS	21 October 2016
Darboux transformations for linear operators on two dimensional regular lattices Authors: Adam Doliwa, M Nieszporski Download PDF	2008	DIS	21 October 2016
Singular finite-gap operators and indefinite metric II$^3$ Authors: Sergey Novikov, P Grinevich Download PDF	2008	DIS	21 October 2016
Desargues maps and the Hirota$-$Miwa equation Authors: Adam Doliwa Download PDF	2008	DIS	21 October 2016
Asymmetric integrable quad-graph equations Authors: Peter Hydon, Claude Michel Viallet Download PDF	2008	DIS	21 October 2016
Coisotropic deformations of algebraic varieties and integrable systems Authors: Boris Konopelchenko, G Ortenzi Download PDF	2008	DIS	21 October 2016
Yang$-$Baxter maps associated to elliptic curves Authors: Vassilios Papageorgiou, AG Tongas Download PDF	2008	DIS	21 October 2016
Multisoliton solutions to the lattice Boussinesq equation Authors: Cheng Zhang, Jarmo Hietarinta Download PDF	2008	DIS	21 October 2016
Difference Picard theorem for meromorphic functions of several variables Authors: R Korhonen Download PDF	2008	DIS	21 October 2016
Non-decreasing functions, exceptional sets and generalised Borel lemmas Authors: RJ Korhonen, Rod Halburd Download PDF	2008	DIS	21 October 2016
Lagrangian multiform structure for the lattice KP system Authors: Frank Nijhoff, Reinout Quispel, Sarah Lobb Download PDF	2008	DIS	21 October 2016
Chebyshev polynomial approximations for some hypergeometric systems Authors: Juri Rappoport Download PDF	2008	DIS	21 October 2016
How to detect the integrability of discrete systems Authors: Rod Halburd, Alfred Ramani, Claude Michel Viallet, B Grammaticos Download PDF	2008	DIS	21 October 2016
Expansions on special solutions of the first $\it q$-Painlevé equation around the infinity Authors: Yousuke Ohyama Download PDF	2008	DIS	21 October 2016
The discrete potential Boussinesq equation and its multisoliton solutions Authors: Kenji Kajiwara, K Maruno Download PDF	2008	DIS	21 October 2016
Analytic solutions and integrability for bilinear recurrences of order six Authors: ANW Hone Download PDF	2008	DIS	21 October 2016
Direct "delay" reductions of the Toda hierarchy Authors: Paul Spicer, Nalini Joshi Download PDF	2008	DIS	21 October 2016
Semi-classical Laguerre polynomials and a third order discrete integrable equation Authors: Frank Nijhoff, Paul Spicer Download PDF	2008	DIS	21 October 2016
Integrable non-QRT mappings of the plane Authors: Nalini Joshi, Pavlos Kassotakis Download PDF	2008	DIS	21 October 2016
Elliptic solutions of ABS lattice equations Authors: James Atkinson, Frank Nijhoff Download PDF	2008	DIS	21 October 2016
A constructive approach to the soliton solutions of integrable quadrilateral lattice equations Authors: Frank Nijhoff, James Atkinson Download PDF	2008	DIS	21 October 2016
Growth of degrees of integrable mappings Authors: Peter van der Kamp Download PDF	2008	DIS	21 October 2016
Okamoto's space for the first Painlevé equation in Boutroux coordinates Authors: Hans Duistermaat, Nalini Joshi Download PDF	2008	DIS	21 October 2016
On the non-multiaffine consistent around the cube lattice equations Authors: Pavlos Kassotakis, M Nieszporski Download PDF	2008	DIS	21 October 2016
The $\widehat{\mathfrak{sl}}(n)_{k}$-WZNW fusion ring: A combinatorial construction and a realisation as quotient of quantum cohomology Authors: Christian Korff, Catharina Stroppel Download PDF	2009	DIS	21 October 2016

Workshops

DISW01

Quantum integrable discrete systems

23 March 2009 to 27 March 2009

Satellite

DISW02

Geometric aspects of discrete and ultra-discrete integrable systems

30 March 2009 to 3 April 2009

Satellite

DISW03

Seminars

Tuesday 27th January 2009
13:30 to 15:30	Informal discussions	Discussion Room

Wednesday 28th January 2009
14:00 to 15:00	Jon Nimmo University of Glasgow Darboux transformations and quasideterminants	Room 1

Tuesday 3rd February 2009
11:00 to 13:00	Informal discussions	Discussion Room
14:00 to 15:00	Informal discussion	Discussion Room

Wednesday 4th February 2009
14:00 to 14:50	Yang-Baxter maps associated to integrable lattice equations	Room 1
15:10 to 16:00	Christian Korff University of Glasgow From quantum to ultra-discrete integrable systems: the six-vertex model and the Takahashi-Satsuma cellular automaton	Room 1

Thursday 5th February 2009
11:00 to 13:00	Informal discussions	Room 1

Wednesday 11th February 2009
14:00 to 14:50	Jarmo Hietarinta University of Turku Bilinearization and soliton solutions for the ABS list	Room 1
15:10 to 15:50	Adam Doliwa University of Warmia and Mazury The tau-function of the quadrilateral lattice as Fredholm determinant within the nonlocal D-bar dressing method	Room 1

Thursday 12th February 2009
11:00 to 12:00	Decio Levi Università degli Studi Roma Tre The generalised symmetry method for discrete equations	Room 1

Wednesday 18th February 2009
14:00 to 14:50	Mourad Ismail University of Central Florida Freud weights and their q-analogues	Room 1
15:10 to 16:00	Yasuhiro Ohta Kobe University Soliton solutions for coupled nonlinear Schr\"odinger equations	Room 1

Tuesday 24th February 2009
11:00 to 13:00	Sergey Novikov University of Maryland, College Park Informal talks	Room 1

Wednesday 25th February 2009
14:00 to 14:50	Peter van der Kamp La Trobe University Periodic reductions of integrable lattice equations; the staircase method	Room 1
15:10 to 16:00	Rafael Hernandez Heredero E.U.I.T de Telecomunicaciones The multiscale method for dispersive partial differential equations and integrability	Room 1

Wednesday 4th March 2009
14:00 to 14:50	James Atkinson University of Sydney Schwarzian integrable systems and the Mobius group: a new connection	Room 1
15:10 to 16:00	Algebra and geometry of Somos sequences	Room 1

Monday 9th March 2009
17:00 to 18:00	Sergey Novikov University of Maryland, College Park New Discretization of Complex Analysis	Room 1

Tuesday 10th March 2009
11:00 to 12:30	Valentin Ovsienko Université Claude Bernard Lyon 1 The Pentagram map: a discrete integrable system	Room 1

Wednesday 11th March 2009
14:00 to 14:50	Pavlos Xenitidis University of Leeds Symmetry reductions of integrable lattice equations	Room 1
15:10 to 16:00	Vassilios Papageorgiou University of Patras Yang-Baxter maps with first degree polynomial Lax matrices	Room 1

Tuesday 17th March 2009
11:00 to 15:00	Informal discussions	Discussion Room

Wednesday 18th March 2009
14:00 to 14:50	Alexander Rasin Bar-Ilan University Infinitely many conservation laws for the discrete KdV equation	Room 1
15:00 to 16:00	Ian Marshall Loughborough University Poisson properties of the Schrödinger equation	Room 1

Thursday 19th March 2009
11:00 to 11:50	Francesco Calogero Università degli Studi di Roma La Sapienza Recent findings on isochronous and integrable dynamical systems: can they somehow be extended to discrete-time evolutions?	Room 1
14:00 to 16:00	Informal discussions	Discussion Room

Friday 20th March 2009
14:00 to 15:00	Chris Ormerod La Trobe University Orthogonal polynomials and connection preserving deformations	Room 1

Monday 30th March 2009
09:00 to 17:00	Conference	No Room Required

Tuesday 31st March 2009
09:00 to 17:00	Conference	No Room Required

Wednesday 1st April 2009
09:00 to 17:00	Conference	No Room Required

Thursday 2nd April 2009
09:00 to 17:00	Conference	No Room Required

Friday 3rd April 2009
09:00 to 17:00	Conference	No Room Required

Wednesday 8th April 2009
14:00 to 14:50	Peter Ellsworth Hydon University of Surrey Conservation laws via the variational difference complex	Room 1

Wednesday 15th April 2009
14:00 to 14:50	Hjalmar Rosengren Chalmers University of Technology Three-coloured chessboards and Baxter's eight-vertex-solid-on-solid model	Room 1

Wednesday 22nd April 2009
14:00 to 14:50	Discrete counterparts of second-order two-dimensional linear differential equations and their Darboux transformations	Room 1
15:10 to 16:00	Peter Clarkson University of Kent Vortices and polynomials	Room 1

Wednesday 29th April 2009
14:00 to 14:50	Christian Mercat Université de Montpellier 2 Linear and Quadratic discrete holomorphic functions as integrable systems	Room 1

Wednesday 6th May 2009
14:00 to 14:50	Hidetaka Sakai University of Tokyo Monodromy preserving deformation and 4-dimensional Painlev\'e type equations	Room 1
15:10 to 16:00	Elizabeth Mansfield University of Kent Moving frames and Noether's theorem	Room 1

Thursday 7th May 2009
11:00 to 13:00	Hans Duistermaat Universiteit Utrecht QRT maps and elliptic surfaces	Room 1

Saturday 9th May 2009
18:00 to 19:00	Wine reception	No Room Required

Monday 18th May 2009
17:00 to 18:00	Hans Duistermaat Universiteit Utrecht QRT maps and elliptic surfaces	Room 1

Tuesday 19th May 2009
13:00 to 15:00	Hans Duistermaat Universiteit Utrecht QRT maps and elliptic surfaces (part 2 of series)	Room 1

Wednesday 20th May 2009
10:00 to 12:30	Yang Shi University of Sydney Special solutions of q-Painleve equations via iso-monodromy problem	Room 1
14:00 to 14:50	Richard Askey University of Wisconsin-Madison The world of q and orthagonal polynomials	Room 1
15:10 to 16:00	Teruhisa Tsuda Kyushu University From KP/UC hierarchies to Painleve equations	Room 1

Thursday 21st May 2009
13:30 to 16:00	Discussion	Discussion Room

Tuesday 26th May 2009
13:00 to 13:50	John Roberts University of New South Wales Symplectic structures for difference equations including integrable maps	Room 1
14:00 to 14:50	Hans Duistermaat Universiteit Utrecht QRT maps and elliptic surfaces (part 3 of series)	Room 1

Wednesday 27th May 2009
14:00 to 15:00	Peter Ellsworth Hydon University of Surrey New integrable quad-graphs	Discussion Room

Tuesday 2nd June 2009
11:00 to 12:30	The elliptic Painleve equation	Room 2

Wednesday 3rd June 2009
11:00 to 12:30	The elliptic Painleve equation	Room 1
14:00 to 14:50	Integrable self-adaptive mesh method for the Camassa-Holm and short pulse equations	Room 1

Friday 5th June 2009
14:00 to 16:00	Tomoyuki Takenawa Tokyo University of Marine Science and Technology Informal talks	Discussion Room

Tuesday 9th June 2009
11:00 to 12:00	Anne Granier Université Paul Sabatier Toulouse III A Galois D-groupoid for q-difference equations	Room 1

Wednesday 10th June 2009
14:00 to 14:50	Igor Krichever Columbia University Abelian solutions of difference equations	Room 1
15:10 to 16:00	Boris Konopelchenko Universita' del Salento Discrete deformations of associative algebras and discrete integrable equations	Room 1

Friday 12th June 2009
11:00 to 11:50	Generalised Kramers-Wannier duality for lattice spin models with non-abelian symmetry	Room 1
14:00 to 15:00	James Atkinson University of Sydney Informal talks	Room 2

Monday 15th June 2009
11:00 to 12:30	The elliptic Painleve equation (part 3)	Room 1

Tuesday 16th June 2009
11:00 to 11:50	Da-jun Zhang Shanghai University Getting more solutions from solitons: a limit approach	Room 1
17:00 to 18:00	Bletchley Park 1942 - 1945	Room 1

Wednesday 17th June 2009
14:00 to 14:50	Alexander Mikhailov University of Leeds From automorphic Lie Algebras to discrete integrable systems	Room 1
15:10 to 16:00	Jean-Marie Maillard Université Pierre et Marie Curie Paris The theory of the 2-d ising model is nothing but the theory of elliptic curves	Room 1

Thursday 18th June 2009
11:00 to 12:00	Evgeny Korotyaev Cardiff University Informal seminar: The Hamiltonians and the action variables for periodic KDV and NLS	Room 1

Friday 19th June 2009
11:00 to 12:00	Boris Konopelchenko Universita' del Salento Informal seminar: On the deformation theory of associative algebras and integrable equations	Room 2

Tuesday 23rd June 2009
14:00 to 14:50	Nevanlinna theory and difference Painleve equations	Room 2

Wednesday 24th June 2009
14:00 to 14:50	On the dynamics of the recurrence $x_{n + 1} = f(x _n)/x_{n-1}$	Room 2
15:10 to 16:00	Alexander Veselov Loughborough University Geometry in DIS	Room 1

Thursday 25th June 2009
14:00 to 15:00	Informal seminar: Somos-6 and multiquadratic pencils	Room 2

Organisers

Rod Halburd University College London
Frank Nijhoff University of Leeds
Masatoshi Noumi Kobe University
Reinout Quispel La Trobe University
Orlando Ragnisco Università degli Studi Roma Tre

Speakers

Richard Askey University of Wisconsin-Madison
James Atkinson University of Sydney
Francesco Calogero Università degli Studi di Roma La Sapienza
Peter Clarkson University of Kent
Adam Doliwa University of Warmia and Mazury
Hans Duistermaat Universiteit Utrecht
Anne Granier Université Paul Sabatier Toulouse III
Rafael Hernandez Heredero E.U.I.T de Telecomunicaciones
Jarmo Hietarinta University of Turku
Peter Ellsworth Hydon University of Surrey
Mourad Ismail University of Central Florida
Boris Konopelchenko Universita' del Salento
Christian Korff University of Glasgow
Evgeny Korotyaev Cardiff University
Igor Krichever Columbia University
Decio Levi Università degli Studi Roma Tre
Jean-Marie Maillard Université Pierre et Marie Curie Paris
Elizabeth Mansfield University of Kent
Ian Marshall Loughborough University
Christian Mercat Université de Montpellier 2
Alexander Mikhailov University of Leeds
Jon Nimmo University of Glasgow
Sergey Novikov University of Maryland, College Park
Yasuhiro Ohta Kobe University
Chris Ormerod La Trobe University
Valentin Ovsienko Université Claude Bernard Lyon 1
Vassilios Papageorgiou University of Patras
Alexander Rasin Bar-Ilan University
John Roberts University of New South Wales
Hjalmar Rosengren Chalmers University of Technology
Hidetaka Sakai University of Tokyo
Yang Shi University of Sydney
Tomoyuki Takenawa Tokyo University of Marine Science and Technology
Teruhisa Tsuda Kyushu University
Peter van der Kamp La Trobe University
Alexander Veselov Loughborough University
Pavlos Xenitidis University of Leeds
Da-jun Zhang Shanghai University

Show All

Programme theme

Preprints

On a nonlinear integrable difference equation on the square 3D-inconsistent

The generalised symmetry method for discrete equations

Soliton solutions for ABS lattice equations: I Cauchy matrix approach

Tzitzeica solitons vs. relativistic Calogero-Moser 3-body clusters

Symmetries and integrability of discrete equations defined on a black-white lattice

Singular finite-gap operators and indefinite metric. I.

Lagrangian multiforms and multidimensional consistency

Integrability, analyticity, isochrony, equilibria, small oscillations and Diophantine relations

Three-dimensional discrete systems of Hirota-Kimura type and deformed Lie-Poisson algebras

Linear quadrilateral lattice equations and multidimensional consistency

Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa$-$Holm type equation

On the Neimark$-$Sacker bifurcation in a discrete predator-prey system

Note on the Poisson structure of the damped oscillator

On the non-integrability of the Popowicz peakon system

On non-standard numerical integration methods for biological oscillators

A self-adaptive mesh method for the Camassa$-$Holm equation

Holomorphic curves with shift-invariant hyperplane preimages

Darboux transformations for linear operators on two dimensional regular lattices

Singular finite-gap operators and indefinite metric II$^3$

Desargues maps and the Hirota$-$Miwa equation

Asymmetric integrable quad-graph equations

Coisotropic deformations of algebraic varieties and integrable systems

Yang$-$Baxter maps associated to elliptic curves

Multisoliton solutions to the lattice Boussinesq equation

Difference Picard theorem for meromorphic functions of several variables

Non-decreasing functions, exceptional sets and generalised Borel lemmas

Lagrangian multiform structure for the lattice KP system

Chebyshev polynomial approximations for some hypergeometric systems

How to detect the integrability of discrete systems

Expansions on special solutions of the first $\it q$-Painlevé equation around the infinity

The discrete potential Boussinesq equation and its multisoliton solutions

Analytic solutions and integrability for bilinear recurrences of order six

Direct "delay" reductions of the Toda hierarchy

Semi-classical Laguerre polynomials and a third order discrete integrable equation

Integrable non-QRT mappings of the plane

Elliptic solutions of ABS lattice equations

A constructive approach to the soliton solutions of integrable quadrilateral lattice equations

Growth of degrees of integrable mappings

Okamoto's space for the first Painlevé equation in Boutroux coordinates

On the non-multiaffine consistent around the cube lattice equations

The $\widehat{\mathfrak{sl}}(n)_{k}$-WZNW fusion ring: A combinatorial construction and a realisation as quotient of quantum cohomology

Workshops

Quantum integrable discrete systems

Geometric aspects of discrete and ultra-discrete integrable systems

Algebraic theory of difference equations

Discrete systems and special functions

Discrete integrable systems - a follow-up meeting

Seminars

Organisers

Speakers

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