# Workshop Programme

## for period 8 - 12 July 2013

### Discrete Integrable Systems - a follow-up meeting

8 - 12 July 2013

Timetable

Monday 8 July | ||||

09:00-09:25 | Registration at INI | |||

09:25-09:30 | Welcome and Opening Remarks by INI Director John Toland | |||

09:30-10:00 | Joshi, N (University of Sydney) |
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Quicksilver solutions of a q-discrete Painlevé equation | Sem 1 | |||

Critical solutions of the classical Painlevé equations arise as universal limits in many nonlinear systems. Their asymptotic properties have been studied from several different points of view. This talk focusses on their discrete versions, for which many questions remain open. Much of the activity in this field has been concentrated on deducing the correct discrete versions of the Painlevé equations, finding transformations and other algebraic properties and describing solutions that can be expressed in terms of earlier known functions, such as q-hypergeometric functions. In this talk, I focus on solutions that cannot be expressed in terms of earlier known functions. In particular, I will describe solutions of the so called q-PI equation, which is a q-discrete versionof the first Painlevé equation. The solutions I will describe are analogous to the critical or the tritronquée solutions, but their complex analytic properties differ. For this reason, I propose a new name: quicksilver solutions and provide a glimpse into their asymptotic properties. |
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10:00-10:30 | Hietarinta, J (University of Turku) |
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Discrete Boussinesq equations | Sem 1 | |||

We discuss lattice versions of the Boussinesq equation. Since the continuous form is not evolutionary (i.e., first order in time derivatives) but second order in time, the Boussinesq equation cannot be discretized as a one-component equation on an elementary quadrilateral of the Cartesian lattice. Instead there are one-component discretizations on a 3 x 3 stencil and three-component versions on the quadrilateral. Furthermore, discrete bilinear versions are also known. We compare these different approaches and the related soliton solutions. |
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10:30-11:00 | Morning Coffee | |||

11:00-11:30 | Schief, W (University of New South Wales) |
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Bianchi (hyper-)cubes and a geometric unification of the Hirota and Miwa equations | Sem 1 | |||

We present a geometric and algebraic way of unifying two discrete master equations of soliton theory, namely the dKP (Hirota) and dBKP (Miwa) equations. We demonstrate that so-called Cox lattices encapsulate Bianchi (hyper-)cubes associated with either simultaneous solutions of a novel 14-point and the dBKP equations or solutions of the dKP equation, depending on whether the Cox lattices are generic or degenerate. |
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11:30-12:00 | Dzhamay, A (University of Northern Colorado) |
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Discrete Schlesinger Transformations and Difference Painlevé Equations | Sem 1 | |||

We study a discrete version of isomonodromic deformations of Fuchsian systems, called Schlesinger transformations, and their reductions to discrete Painlevé equations. We obtain an explicit formula for the generating function of elementary Schlesinger transformations in terms of the coordinates on the so-called decomposition space associated to the Fuchsian system and interpret it as a discrete Hamiltonian of our dynamic. We then consider some explicit examples of reductions of such transformations to discrete Painlevé equations. Using the birational geometry of rational surfaces associated to these equations, we compare the form of the equations that correspond to the elementary Schlesinger transformations to standard form of the equations of the same type. |
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12:15-13:30 | Lunch at Wolfson Court | |||

14:00-14:30 | Zhang, DJ (Shanghai University) |
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The Sylvester equation, Cauchy matrices and matrix discrete integrable systems | Sem 1 | |||

The Sylvester equation, AM+ MB = C, is a famous matrix equation in linear algebra and widely used in many areas. Solution (M) of the equation is usually given through matrix exponential functions and integrals. In the talk, we will give an explicit form of M as solutions to the Sylvester equation. Then, starting from the Sylvester equation and introducing suitable shift relations to define plane wave factors, we construct matrix discrete integrable systems of which solutions are explicitly expressed via Cauchy matrices. |
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14:30-15:00 | Yoo-Kong, S (King Mongkut's University of Technology Thonburi) |
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The variational principle for Lagrangian 1-forms | Sem 1 | |||

We present the recently proposed variational principle for Lagrangian 1-form structures both discrete and continuous in the space of multi-time variables, in which the action functionals depend not only on the dynamical (dependent) variables of the system but also on the (parametrised) curves in the space of (independent) time-variables which define the dynamics in terms of multitime fl ows. It was conjectured that the only admissible Lagrangians allowing such structures are the integrable ones. |
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15:00-15:30 | Afternoon Tea | |||

15:30-16:00 | Nieszporski, M (Uniwersytet Warszawski) |
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Reductions of DIS. Discrete Painlevé equations and Painlevé correspondences | Sem 1 | |||

With many of difference equations one can associate a superior discrete integrable system closely related to Yang-Baxter maps. This association turns out to be helpful in understanding some correspondences (multivalued recurrences) that are consistent-around-the-cube. Moreover, the standard procedure of periodic reductions leads not only to Painlevé equations but also to their multivalued generalizations. |
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16:00-16:30 | Atkinson, J (University of Sydney) |
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Fano 3-space and symmetrisation of quad-equations | Sem 1 | |||

Recent results have shown how invariance under interchange of variables and parameters of integrable discrete models can lead to significant generalisation of the domain; giving a more natural lattice geometry that respects the symmetry. I will describe a symmetric generalisation of the polynomials that define the principal rational, trigonometric and elliptic models from the ABS list, namely Q2, Q3 and Q4. They have a property, generalising the consistency on the cube, which involves the finite geometry PG(3,2). This space was discovered originally during investigation of the axiomatic framework for projective geometry by Gino Fano in 1892, and is connected with symmetrisation of the quadrangle. |
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16:30-17:30 | Welcome Wine Reception |

Tuesday 9 July | ||||

09:30-10:00 | Calogero, F (Università degli Studi di Roma La Sapienza) |
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New solvable discrete-time many-body problems featuring several arbitrary parameters | Sem 1 | |||

A technique will be illustrated to identify discrete-time `many-body problems'. The models thereby manufactured feature several parameters and are solvable by algebraic operations. In some cases (i.e., for some subsets of parameters) the solutions are isochronous, i.e. periodic with a fixed period independent of the initial data. This is joint work with F. Leyvraz. |
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10:00-10:30 | Yamada, Y (Kobe University) |
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The q-Painlevé equations arising from the q-interpolation problems | Sem 1 | |||

For the polynomials P(x), Q(x) obtained by a Padé (or Chauchy-Jacobi) interpolation: Y (xi) = P(xi)=Q(xi), we consider the contiguity relations satisfied by the functions P(x) and Y (x)Q(x). In a suitable setup of the interpolation problem, the contiguity relations can be interpreted as a Lax pair for a discrete Painlevé equation. In this sense, the Padé interpolation order a cheap way to get a Lax pair of discrete Painlevé equations together with their special solutions. In this talk, I will discuss this method in some examples of the q-Painlevé equations. |
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10:30-11:00 | Morning Coffee | |||

11:00-11:30 | Konstantinou-Rizos, S (University of Leeds) |
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Darboux transformations, discrete integrable systems and related Yang-Baxter maps | Sem 1 | |||

In this talk we derive Darboux transformations which are invariant under the action of finite reduction groups. We present Darboux transformations for the NLS equation, the derivative NLS equation and a deformation of the derivative NLS equation. We use the associated Darboux matrices to define discrete Lax pairs and derive discrete integrable systems. Moreover, we use these Darboux matrices to construct 6-dimensional Yang-Baxter maps which can be restricted to 4-dimensional YB maps on invariant leaves. The former are completely integrable discrete maps. |
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11:30-12:00 | Doliwa, A (University of Warmia and Mazury) |
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Some non-commutative integrable systems from Desargues maps | Sem 1 | |||

We investigate periodic reductions of Desargues maps, which lead to novel integrable multicomponent lattice systems being non-commutative, non-isosectral, and non-autonomous analogues of the modified Gel'fand Dikii hierarchy. The equations are multidimensionally consistent, and we present the corresponding geometric systems of Lax pairs. We clarify the origin and appearance of functions of single variables, whose presence is indispensable in making further reductions to lattice Painlevé equations. |
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12:15-13:30 | Lunch at Wolfson Court | |||

14:00-14:30 | Hone, A (University of Kent) |
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Cluster algebras and discrete integrable systems | Sem 1 | |||

We consider a large family of nonlinear rational recurrence relations which arise from mutations in cluster algebras defined by quivers. The advantage of the cluster algebra formalism is that it immediately provides an invariant symplectic (or presymplectic) structure. The problem of determining which of the recurrences are integrable in the sense of Liouville's theorem is related to the notion of algebraic entropy, and via a series of conjectures related to tropical (max-plus) algebra, this leads to a very sharp criterion for the allowed degrees of the terms in the recurrence. As a result, four infinite families of discrete integrable systems are obtained. This is joint work with Allan Fordy. |
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14:30-15:00 | Fordy, A (University of Leeds) |
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Integrable maps which preserve functions with symmetries | Sem 1 | |||

We consider maps which preserve functions which are built out of the invariants of some simple vector fields. We give a reduction procedure, which can be used to derive commuting maps of the plane, which preserve the same symplectic form and first integral. We show how our method can be applied to some maps which have recently appeared in the context of Yang-Baxter maps. Based on the paper: A.P. Fordy, P. Kassotakis, Integrable Maps which Preserve Functions with Symmetries, J Phys A: v46, 205201 (2013) |
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15:00-15:30 | Afternoon Tea | |||

15:30-16:00 | Emsiz, E (Pontificia Universidad Católica de Chile) |
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Discrete harmonic analysis on a Weyl alcove | Sem 1 | |||

I will speak about recent work on a unitary representation of the affine Hecke algebra given by discrete difference-reflection operators acting in a Hilbert space of complex functions on the weight lattice of a reduced crystallographic root system. I will indicate why the action of the center under this representation is diagonal on the basis of Macdonald spherical functions (also referred to as generalized Hall-Littlewood polynomials associated with root systems). I will furthermore discuss a periodic counterpart of the above mentioned model that is related to a representation of the double affine Hecke algebra at critical level q = 1 in terms of difference-reflection operators. We use this representation to construct an explicit integrable discrete Laplacian on the Weyl alcove corresponding to an element in the center. The Bethe Ansatz method is employed to show that our discrete Laplacian and its commuting integrals are diagonalized by a finite-dimensional basis of periodic Macdonald spherical functions. This is joint work in progress with J. F. van Diejen. |
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16:00-16:30 | Levi, D (Università degli Studi Roma Tre) |
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Multiscale reductions and integrability on the lattice | Sem 1 | |||

We consider the classification up to a Mbius transformation of multilinear real integrable partial difference equations with dispersion defined on a square lattice by the multiscale reduction around their harmonic solution. We show that the A1, A2, and A3 integrability conditions constrain the number of parameters in the equation. The A4 integrability conditions provide no further constrain suggesting that the obtained equations be integrable. |

Wednesday 10 July | ||||

09:30-10:00 | Nijhoff, F (University of Leeds) |
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Lagrangian multiform theory and variational principle for integrable systems | Sem 1 | |||

During the DIS programme of 2009 Lobb and Nijhoff introduced a novel point of view on the Lagrangian theory of systems integrable in the sense of multidimensional consistency. The key observation was that suitably chosen Lagrangians obey a "closure" relation when embedded in multidimensional, discrete or continuous, space-time and subject to the equations of the motion. The apparent universality of this phenomenon has now been confirmed for many integrable systems, both continuous and discrete with defining equations in one, two and three dimensions. From a physics point of view this could set a new paradigm for least-action principles in physics where the Lagrangian itself is a solution of a system of generalized Euler-Lagrange equations, and where the geometry in the embedding space is a variational variable together with the field variables. |
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10:00-10:30 | Lobb, S (University of Sydney) |
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Variational principle for discrete 2d integrable systems | Sem 1 | |||

For multidimensionally consistent systems we can consider the Lagrangian as a form, closed on the multidimensional equations of motion. For 2-d systems this allows us to define an action on a surface embedded in higher dimensions. It is then natural to propose that the system should be derived from a variational principle which includes not only variations with respect to the dependent variables, but also variations of the surface in the space of independent variables. I will describe how this puts constraints on the Lagrangian, and how this leads to equations on a single quad in the lattice. |
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10:30-11:00 | Morning Coffee | |||

11:00-11:30 | Suris, YB (Technische Universität Berlin) |
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What is integrability of discrete variational systems? | Sem 1 | |||

In this talk, we will address several fundamental aspects of the Lagrangian theory of discrete integrable (multi-dimensionally consistent) systems, which was pioneered in 2009 by Lobb and Nijhoff. |
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11:30-12:00 | Kajiwara, K (Kyushu University) |
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Discrete mKdV flow on discrete space curves | Sem 1 | |||

We formulate the isoperimetric motion of the discrete curves in $\mathbb{R}^3$ described by the discrete mKdV equation. Not only formulating the motion on the level of the Frenet frame determined by the Lax pair, we "integrate" it to give the explicit formula on the level of the curve itself. This work has been done in collaboration with J. Inoguchi, N. Matsuura and Y. Ohta. |
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12:15-13:30 | Lunch at Wolfson Court | |||

14:00-17:00 | Free time followed by Conference Dinner | |||

19:30-22:00 | Conference Dinner at Emmanuel College |

Thursday 11 July | ||||

09:30-10:00 | Roberts, J (University of New South Wales) |
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Exact results for degree growth of lattice equations and their signature over finite fields | Sem 1 | |||

In the first part of the talk, we study the growth of degrees (algebraic entropy) of certain multi-affine quad-rule lattice equations with corner boundary conditions. We work projectively with a free parameter in the boundary values, so that at each vertex, there are 2 polynomials in this parameter. We show the ambient growth of their degree is known exactly, via the asymptotics of the Delannoy double sequence. Then we give a conjectured growth for the degrees of the greatest common divisor that is cancelled at each vertex. Taken together, these provide us with a constant coefficient linear partial difference equation that determines the growth in the reduced degrees at each vertex. For a whole class of equations, including most of the ABS list, this proves polynomial growth of degree. For other equations where the cancellation at each vertex is not high enough, we prove exponential growth. In the second part of the talk, we study integrable lattice equations and their perturbations over finite fields. We discuss some tests that can distinguish between integrable equations and their non-integrable perturbations, and their limitations. Some integrable candidates found using these tests can then be shown to have vanishing entropy via the results of the first part of the talk. Both parts of the talk are joint work with Dinh Tran (UNSW). |
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10:00-10:30 | Kanki, M (University of Tokyo) |
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Integrability of discrete systems over finite fields | Sem 1 | |||

Discrete integrable systems defined over finite fields are studied. Dynamical systems over non-Archimedean fields are of great interest in the theory of arithmetic dynamical systems [1].Discrete integrable equations over the field of p -adic numbers are defined and then the evolutions are reduced to the finite field. The integrable systems are shown to have a property that resembles a 'good reduction' modulo a prime [2]. We observe that this generalization of the good reduction can be used to test integrability of discrete equations over finite fields. We discuss the relation of our methods to other integrability tests, in particular, the 'singularity confinement test'. Applications of our approach to the two-dimensional lattice systems such as the discrete KdV equation are also studied. [1] J. H. Silverman, The Arithmetic of Dynamical Systems, (2007), Springer-Verlag, New York. [2] M. Kanki, J. Mada, K. M. Tamizhmani, T. Tokihiro, Discrete Painleve II equation over finite fields, J. Phys. A: Math. Theor., 45, (2012), 342001 (8pp). |
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10:30-11:00 | Morning Coffee | |||

11:00-11:30 | van der Kamp, P (La Trobe University) |
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Initial value problems for Discrete Integrable Systems | Sem 1 | |||

Initial value problems for quad graphs. We describe a method to construct well-posed initial value problems for not necessarily integrable equations on not necessarily simply connected quad-graphs. Although the method does not always provide a well-posed initial value problem (not all quad-graphs admit well-posed initial value problems) it is effective in the class of rhombic embeddable quad-graphs. |
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11:30-12:00 | van Diejen, JF (Ponticia Universidad Católica de Chile) |
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The semi-infinite q-boson system with boundary interaction | Sem 1 | |||

The q-boson system is a lattice discretization of the one-dimensional quantum nonlinear Schrödinger equation built of particle creation and annihilation operators representing the q-oscillator algebra. Its n-particle eigenfunctions are given by Hall-Littlewood functions. I will discuss a system of q-bosons on the semi-infinite lattice with boundary interactions arising from a quadratic deformation of the q-boson field algebra at the end point and show that the Bethe Ansatz eigenfunctions are given by Macdonald's three-parameter Hall-Littlewood functions with hyperoctahedral symmetry associated with the BC-type root system. From a stationary phase analysis, it then follows that the n-particle scattering matrix factorizes as a product of explicitly computed two-particle bulk and one-particle boundary scattering matrices. |
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12:15-13:30 | Lunch at Wolfson Court | |||

14:00-14:30 | Mikhailov, A (University of Leeds) |
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Formal diagonalisation of the Lax-Darboux scheme and conservation laws of integrable partial differential, differential-difference and partial difference | Sem 1 | |||

Formal diagonalisation of Lax operators leads to formal diagonalisation of the corresponding Darboux transformations and vice versa. The latter enables us to find recurrent relations for generating conservation laws and establish natural relations between the canonical series of local conservation laws for partial differential, differential-difference and partial difference equations. In particular we show that the canonical densities of conservation laws for the symmetries of partial difference equations are also conserved densities for the partial difference equations themselves. |
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14:30-15:00 | Wang, JP (University of Kent) |
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Discrete integrable equations with Dihedral reduction group | Sem 1 | |||

In this talk, we'll present n-dimensional discrete integrable equations with Dihedral reduction group and construct their generalised symmetries and conservation laws. We also demonstrate the link with a Darboux transformation of the related Lax representation. This is a joint work with G. Papamikos and A.V. Mikhailov. |
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15:00-15:30 | Afternoon Tea | |||

15:30-16:00 | Demskoy, D (Charles Sturt University) |
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Quad equations and auto-Bäcklund transformations of NLS-type systems | Sem 1 | |||

Starting with a hierarchy consisting of a quad equation and semi-discrete higher order flows, we show how one can systematically construct related auto-Bäcklund transformations and superposition formulas for solutions of NLS-type systems. |
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16:00-16:30 | Kouloukas, T (University of Patras) |
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Integrable refactorization systems and Yang-Baxter maps | Sem 1 | |||

We present matrix refactorization problems related to Lax representations of Yang-Baxter maps. First degree polynomial Lax matrices can be considered as basic building blogs of higher dimensional Yang-Baxter maps. Initial value problems on lattices give rise to families of Poisson maps which preserve the spectrum of their monodromy matrix. |

Friday 12 July | ||||

09:30-10:00 | Ormerod, C (La Trobe University) |
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Special solutions of the additive discrete Painlevé equation with $E_6^{(1)}$ symmetry | Sem 1 | |||

We consider a reduction of the lattice potential Korteweg-de Vries equation. Using a parameterization of the Lax pair of Arinkin and Borodin, we identify the reduction as the additive discrete Painlevé equation admitting an affine Weyl group of type $E_6^{(1)}$ as a group of Bäcklund transformations. We use the reduced equations to obtain a family of explicitrational and hypergeometricsolutions of this discrete Painlevé equation. |
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10:00-10:30 | Tamizhmani, M (Pondicherry University) |
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Linearisable mappings and their explicit integration | Sem 1 | |||

After a short introduction concerning the presence of linearisable systems in the Painlevé and Gambier classications we shall present recent results on linearisable mappings. The study of discrete linearisable second-order mappings revealed the existence of three main families of such systems. The first ones are the projective mappings, the second group are the linearisable mappings of Gambier type and their various degenerate forms and third family is known under the name of linearisable mapping of third kind. We shall present the linearisable mappings belonging to these families, derive their nonautonomous forms and give their explicit integration. |
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10:30-11:00 | Morning Coffee | |||

11:00-11:30 | Xenitidis, P (University of Leeds) |
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Integrability conditions for difference equations | Sem 1 | |||

The existence of an infinite hierarchy of symmetries is used as a definition for the integrability of difference equations. Using this definition and introducing the notion of a formal recursion operator, we derive some easily verifiable conditions which are necessary for the integrability of a given difference equation. We refer to these relations as integrability conditions and we show how symmetries and conservation laws can be derived from them. Employing these conditions, we establish the integrability of a particular equation and construct its symmetries. Finally, we derive a Miura transformation which maps these symmetries to a generalized Bogoyavlensky lattice. |
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11:30-12:00 | Maruno, K (University of Texas-Pan American) |
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Discrete integrable systems with self-adaptive moving mesh | Sem 1 | |||

In recent years, we have investigated integrable discretizations of nonlinear partial differential equations in which singularities of solutions exist. For PDEs in this class, we obtained several discrete integrable systems with self-adaptive moving mesh which can be used for numerical simulations of nonlinear PDEs. Both the Hirota's discretization method and a geometric method are effective to construct self-adaptive moving mesh discrete integrable systems. In this talk, I will review recent our results of discrete integrable systems with self-adaptive moving mesh. This is joint work with Bao-Feng Feng and Yasuhiro Ohta. |
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12:15-13:30 | Lunch at Wolfson Court | |||

14:00-14:30 | van Assche, W (Katholieke Universiteit Leuven) |
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Discrete Painlevé equations and orthogonal polynomials | Sem 1 | |||

It is now well known that the recurrence coefficients of many semi-classical orthogonal polynomials satisfy discrete and continuous Painlevé equations. In the talk we show how to find the discrete Painlevé equations by using compatibility relations or ladder operators. A combination with Toda type equations allows to find continuous Painlevé equations. We have made an attempt to go through the literature and to collect all known examples of discrete (and continuous) Painlevé equations for the recurrence coefficients of orthogonal polynomials. All input in completing the list is welcome. The required solutions usually satisfy some positivity constraint, leading to a unique solution with one boundary condition. Often the solution corresponds to solutions of the (continuous) Painlevé equation in terms of special functions. |
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14:30-15:00 | Quispel, R (La Trobe University) |
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Some recent results on the Kahan-Hirota-Kimura discretization | Sem 1 | |||

We show that Kahan's discretization of quadratic vector fields is equivalent to a Runge- Kutta method. In case the vector field is Hamiltonian, with constant Poisson structure, the map determined by this discretization preserves a (modified) integral and a (modified) invariant measure. This produces large classes of integrable rational mappings, explaining some of the integrable cases that were previously known, as well as yielding many new ones. |