19-23 June 2023

Faculty of Physics, University of Warsaw

Europe/Warsaw timezone

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58

Session:
Integrable birational maps

When completely integrable Hamiltonian systems are discretised, the resulting discrete-time systems are often no longer integrable themselves. This is the so-called "problem of integrable discretisation". Two known exceptions to this situation in 3D are the Kahan discretisations of the Euler top and the Zhukovski-Volterra gyrostat with one non-zero linear parameter $\beta$, both of degree 3. The i
... More

Presented by Dr. Jaume ALONSO
on
22 Jun 2023
at
15:30

Session:
Discrete PainlevÃ©

For the q-PainlevÃ© equation with affine Weyl group symmetry of type E_6^{(1)}, a 2Ã—2 matrix Lax form and a second order scalar lax form were known.
In this talk, we give a 3Ã—3 matrix Lax form and a third order scalar equation related to it. They seems to be new.

Presented by Ms. Kanam PARK
on
19 Jun 2023
at
18:00

Session:
Non-linear waves

We report that a class of integrable discrete holomorphic functions can generate planar truss structures with a certain mechanical optimality called the Michell structure, well-known in the area of architecture. Further, the discrete planar curves formed by the edges are nothing but the discrete analogue of the logarithmic spiral which is a special case of the discrete log-aesthetic curves. Discr
... More

Presented by Prof. Kenji KAJIWARA
on
21 Jun 2023
at
10:15

Session:
Continuous integrable systems

We extend Fibonacci numbers with arbitrary weights and generalize a dozen Fibonacci identities. As a special case, we propose an elliptic extension which extends the $q$-Fibonacci polynomials appearing in Schur's work. The proofs of most of the identities are combinatorial, extending the proofs given by Benjamin and Quinn, and in the $q$ case, by Garrett. Some identities are proved by telescoping
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Presented by Ms. Archna KUMARI
on
23 Jun 2023
at
12:30

The alternating sign matrices-descending plane partitions (ASM-DPP) bijection problem is one of the most intriguing open problems in integrable combinatorics. Recently, Fischer and Konvalinka have obtained a bijection between ASM(n) Ã— DPP(nâˆ’1) and DPP(n) Ã— ASM(nâˆ’1) using the notions of a signed set and a signed bijection and which involves an explicit construction of a signed bijection betwe
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Presented by Mr. Takuya INOUE

Session:
Non-commutative

We propose to revisit the problem of quantisation and look at it from an
entirely new angle, focussing on quantisation of dynamical systems themself, rather than of their Poisson structures. We begin with a lift of a classical dynamical system to a system on a free associative algebra with noncommutative dynamical variables and reduce the problem of quantisation
to the problem of studying of two
... More

Presented by Prof. Alexander MIKHAILOV
on
20 Jun 2023
at
09:15

Session:
Ultra discrete equations

In this talk we look at some simple visual methods to analyse the
solutions to ultra-discrete equations with real values for the
dependent variable as opposed to integer values.
Using these visual techniques the number of solitons present can be
determined and solitons can be easily removed or added to the system.

Presented by Dr. Claire GILSON
on
20 Jun 2023
at
15:00

Session:
Discrete differential geometry

Presented by Prof. Alexander BOBENKO
on
20 Jun 2023
at
17:30

Session:
Discrete PainlevÃ©

The Garnier system is an extension of the sixth Painlev\'e equation from a viewpoint of the isomonodromy deformation of a Fuchsian system.
Its $q$-difference analogue was proposed by Sakai as the connection preserving deformation of a linear $q$-difference system.
Recently, we formulated the $q$-Garnier system in a framework of an extended affine Weyl group of type $A^{(1)}_{2n+1}\times A^{(1)}_
... More

Presented by Prof. Takao SUZUKI
on
19 Jun 2023
at
17:00

Session:
Darboux polynomials

Celledoni et al [1,2] have illustrated the power of using so-called Darboux polynomials to search for preserved integrals and measures of a rational map L (including a map arising as a Kahan-Hirota-Kimura discretisation of a (Hamiltonian) ODE). For instance, integrals are obtained as the ratio of two Darboux polynomials where each Darboux polynomial satisfies an equation of the form P(x') = C(x)
... More

Presented by Prof. John ROBERTS
on
22 Jun 2023
at
17:00

Session:
Ultra discrete equations

We give positive answers to some conjectures due to Ismail contained in his monograph of 2005. This concerns characterizations theorems for orthgonal polynomials on lattices.

Presented by Dr. Dieudonne MBOUNA
on
20 Jun 2023
at
15:40

Session:
Continuous integrable systems

Dupin cyclidic (DC) coordinates in R^3 are triple orthogonal coordinates where all coordinate lines are circles or straight lines. Besides classical examples (spherical, cylindrical, conical, etc.) there are less-known DC coordinates constructed using two focal conics (ellipse and hyperbola, or two parabolas) in orthogonal planes, used by Darboux for separation of variables in the Laplace equation
... More

Presented by Dr. Rimvydas KRASAUSKAS
on
23 Jun 2023
at
17:00

We introduce the concept of coalgebra symmetry for discrete systems. We use this powerful tool to prove (super)integrability, superintegrability of some discrete systems in arbitrary dimension. In particular, we study the systems obtained by the coalgebra symmetry with respect to special linear Lie--Poisson algebra in dimension 2.

Presented by Dr. Giorgio GUBBIOTTI

One of the most important dynamical invariant associated to a birational map f is given by its dynamical degree, or equivalently, its algebraic entropy, which is defined via the rate of growth of the sequence deg(f^n). More concretely, the degrees deg(f^n), although not birationally invariant by themselves, are also of great interests in understanding the dynamics of the birational map.
We propo
... More

Presented by Mr. Kangning WEI
on
22 Jun 2023
at
12:30

Session:
Compatibility

In this talk, we present some details of how to implement the consistency scheme for 5-point lattice equations broadly introduced in the talk by Wolfgang Schief. We will also present some other formulations of consistency for 5-point lattice equations, particularly for equations in a hexagonal lattice. Time permitting we will also present consistent multicomponent 5-point equations and consisten
... More

Presented by Dr. Andrew KELS
on
19 Jun 2023
at
10:00

Session:
Compatibility

Consistency of discrete equations on higher dimensional lattices constitutes a central element of integrable systems theory. The consistency of discrete equations defined on the squares and cubes of lattices of type B and the octahedra of lattices of type A have been studied extensively and with great success. However, it appears that the consistency of discrete equations naturally defined on latt
... More

Presented by Prof. Wolfgang SCHIEF
on
19 Jun 2023
at
09:30

Session:
Darboux polynomials

We will discuss (mainly linear) Darboux polynomials for ODEs and for difference equations, and show their relation to the preservation of measures and first integrals, and in the construction of integrable systems.

Presented by Prof. Reinout QUISPEL
on
22 Jun 2023
at
16:30

Session:
Cluster algebras

We present parametric deformations of sequences of cluster mutations in the framework of cluster algebras, which destroy the Laurent property but preserve a presymplectic structure induced by the corresponding exchange matrix. We investigate the Liouville integrability of deformed parametric cluster maps associated with the A3 and A4 quivers by imposing suitable constraints on the parameters. We
... More

Presented by Dr. Theodoros KOULOUKAS
on
19 Jun 2023
at
15:00

Session:
.

The (quantum) van Diejen model is an integrable many-body system defined by a family of mutually commuting analytic difference operators that is known to have hyperoctahedral symmetry in its variables and $E_8$ Weyl group symmetry in its parameters (under certain constraint).
In this talk, new generalizations of the van Diejen Hamiltonian - and some exact eigenfunctions - are presented. In parti
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Presented by Farrokh ATAI
on
20 Jun 2023
at
11:30

Integrability criteria that have been enormously successful for second order mappings, such as singularity confinement or zero algebraic entropy, are often applied to lattice equations as though the latter were mere mappings.
In this talk we will show that such a naÃ¯ve approach can (and does) lead to all sorts of contradictions and that considerable care is needed when using such methods to inve
... More

Presented by Dr. Takafumi MASE
on
22 Jun 2023
at
11:30

We study the growth of complexity, or degree growth, of one-component
lattice equations defined on a 3x3 stencil. The 2x2 case was discussed
in a previous talk by T. Mase. The equations studied here include two
7-point equations in Hirota bilinear form as well as 9-point
Boussinesq equations of regular, modified and Schwarzian type.
The initial values are given on a staircase or on a corner
... More

Presented by Prof. Jarmo HIETARINTA
on
22 Jun 2023
at
12:00

It has been known for some years that there are deep connections between continued fractions and integrable systems: one of the earliest examples appears in Moser's work on solutions of the finite Kac-van Moerbeke (or Volterra) lattice, but there are many other examples e.g. in recurrence coefficients for orthogonal polynomials arising in random matrix theory, which satisfy (discrete and continuou
... More

Presented by Prof. Andrew HONE
on
23 Jun 2023
at
09:15

I will present certain birational maps that are solutions of the parametric entwining
Yang-Baxter equation. These maps are obtained via the refactorisation problem of
certain Darboux transformations associated with the Lax operators of certain soliton
PDEs. I will also present various dynamical properties of the derived maps, such
as existence of invariants and associated symplectic or Poisson
... More

Presented by Dr. Georgios PAPAMIKOS
on
19 Jun 2023
at
12:00

Session:
Continuous integrable systems

We extend the expansion formulas of Liu given in 2013 to the context of multiple series over root systems. Liu and others have shown the usefulness of these formulas in Special Functions and number-theoretic contexts. We extend Wang and Maâ€™s generalizations of Liuâ€™s work which they obtained using q-Lagrange inversion. We use the A_n and C_n Bailey transformation and other summation theorems du
... More

Presented by Ms. SURBHI RAI
on
23 Jun 2023
at
12:00

Session:
Non-commutative

The term "K-theoretic symmetric function" refers to a family of symmetric functions representing Schubert varieties in the K-theory of flag varieties. (They are referred to as "Grothendieck polynomials" for type A and as "K-Q-functions" for type C.) In this talk, I will introduce a fermionic description of these K-theoretic symmetric functions in terms of the boson-fermion correspondence. This met
... More

Presented by Dr. Shinsuke IWAO
on
20 Jun 2023
at
10:15

Session:
Singularity analysis

Since its introduction, the method of full deautonomisation by singularity confinement has proved a strikingly effective way of detecting the dynamical degrees of birational mappings of the plane.
This method is based on a conjectured link between two a priori unrelated notions: firstly the dynamical degree of the mapping and secondly the evolution of parameters required for its singularity struc
... More

Presented by Dr. Alexander STOKES
on
22 Jun 2023
at
09:45

Session:
Darboux polynomials

The motivation of our research is to find structure preserving discretizations of dynamical systems by developing some ideas related to the method of discrete gradients [1].
First, we recall that the discrete gradient method can be improved in two different ways without losing the energy conservation property, either by increasing its order [2], or by so-called locally exact discretizations that
... More

Presented by Jan CIEÅ›LIÅ„SKI
on
22 Jun 2023
at
18:00

We will present a geometrical interpretation of the Lagrangian 1-form closure relation, inferred as the multi-dimensional consistency in time evolution on the space of independent variables. New mathematical objects, such as Lagrange vector field and Hamilton vector field defined on the the space of independent variables, will be used to capture the integrability condition.

Presented by Dr. Sikarin YOO-KONG

Session:
Integrable birational maps

Motivated by the study of the Kahan--Hirota--Kimura discretisation of the Euler top, we characterise the growth and integrability properties of a collection of elements in the Cremona group of a complex projective 3-space using techniques from algebraic geometry. This collection consists of maps obtained by composing the standard Cremona transformation in three dimensions with projectivities that
... More

Presented by Dr. Giorgio GUBBIOTTI
on
22 Jun 2023
at
15:00

Session:
Non-commutative

Integrable nonabelian systems are equations of motion in which the field variables take values in a noncommutative algebra, as a matrix one. In a series of papers with Jing Ping Wang (Nonlinearity 2021, CMP 2022) we have investigated the Hamiltonian structure and recursion operators for hierarchies of differential-difference integrable equations, providing a geometrical interpretation that helps t
... More

Presented by Dr. Matteo CASATI
on
20 Jun 2023
at
09:45

Session:
Discrete differential geometry

We consider a classical problem in differential geometry, known as the Bonnet problem, whether a surface is characterized by a metric and mean curvature function. We explicitly construct a pair of immersed tori that are related by a mean curvature preserving isometry. This resolves a longstanding open problem on whether the metric and mean curvature function determine a unique compact surface. Dis
... More

Presented by Prof. Alexander BOBENKO
on
20 Jun 2023
at
17:00

The discrete integrable structure concealed by orthogonal matrix ensembles is the Pfaff lattice, built on a semi-infinite moment matrix defined for a skew-symmetric weight. The dependence on the infinitely many times is encoded at the level of the weight, and gives rise to an integrable hierarchy expressed as the collection of infinitely many Lax equations. The leading order of the continuum limit
... More

Presented by Dr. Marta DELL'ATTI

Session:
.

The notion of boundary conditions for quad-graph systems will first be introduced. The boundary conditions are naturally defined on triangles that arise as dualization of given quad-graphs with boundary. For three-dimensionally consistent quad-graph systems, the so-called integrable boundary conditions will be characterized as boundary conditions satisfying the boundary consistency condition that
... More

Presented by Dr. Cheng ZHANG
on
20 Jun 2023
at
12:00

Session:
Cluster algebras

An integrable deformation of a cluster map is an integrable Poisson map which is composed of a sequence of deformed cluster mutations, namely, parametric birational maps preserving the presymplectic form but destroying the Laurent property, which is a necessary part of the structure of a cluster algebra. However, this does not imply that the deformed map does not arise from a cluster map: one can
... More

Presented by Mr. Wookyung KIM
on
19 Jun 2023
at
15:30

Session:
.

Integrable maps can be obtained as periodic or open reductions from lattice equations which satisfy consistency conditions.
An open reduction (this will be explained in the talk by Cheng Zhang) from Q1_0 was decomposed as a composition of Manin involutions on each curve of the invariant pencil.
However, one of the involution points appeared to be curve-dependent....

Presented by Dr. Peter VAN DER KAMP
on
20 Jun 2023
at
12:30

Session:
Non-linear waves

2D-Toda lattices corresponding to the Cartan matrices of simple Lie algebras are known to be Darboux integrable, i.e. they admit complete families of essentially independent characteristic integrals. During the last three decades various discrete analogs of these systems were obtained. In 2011 Habibullin proposed a systematic way to discretize 2D-Toda lattices. His approach was based on the
idea
... More

Presented by Dr. Sergey SMIRNOV
on
21 Jun 2023
at
09:45

Session:
Darboux polynomials

The Kahan discretization of a Lotka-Volterra system leads to a rational map parametrized by the step size. When this map is Poisson with respect to the quadratic Poisson structure of the Lotka-Volterra system we say that this system has the Kahan-Poisson property. There is a well known family of Lotka-Volterra systems having the Kahan-Poisson property. Their underlying graph has n vertices 1, 2, .
... More

Presented by Dr. Charalampos EVRIPIDOU
on
22 Jun 2023
at
17:30

Session:
Lagrangian multiform theory

Recently I presented a Lagrangian 3-form structure for a generalised Darboux system. The original Darboux system arose in connection with the theory of conjugate nets for systems of orthogonal curvilinear coordinates. The generalised system, in which the relevant fields are labelled by continuous parameters which can be associated with lattice parameters of an underlying
3-dimensional integrable
... More

Presented by Prof. Frank NIJHOFF
on
21 Jun 2023
at
11:30

Session:
Continuous integrable systems

In the present talk we address a longstanding problem of search for integrable partial differential systems in four independent variables, i.e. in the case most relevant for possible applications in physics, and show that such systems are significantly less exceptional than it appeared before: in addition to a number of previously known examples like the (anti)self-dual Yang--Mills equations there
... More

Presented by Prof. Artur SERGYEYEV
on
23 Jun 2023
at
17:20

Session:
Lagrangian multiform theory

In aiming to find simple nonlinear Lagrangian 1-forms that exhibit all the relevant features of a multiform structure, we investigate Lagrange structures arising from addition formulae. This results in non-quadratic Lagrangians and commuting maps with discrete and continuous interpolation flows. We discuss the relationship to integrable quad equations and applications to quantum propagators.

Presented by Mr. Jacob RICHARDSON
on
21 Jun 2023
at
12:00

Session:
Continuous integrable systems

The quasi-Painleve property of a system of ordinary differential
equations, here meaning the condition that movable singularities reachable by analytic continuation along finite length curves are at worst algebraic branch points, is described in terms of global Hamiltonian structures on an analogue of Okamotoâ€™s spaces of initial conditions for the Painleve equations. This is a joint work with
... More

Presented by Dr. Galina FILIPUK
on
23 Jun 2023
at
11:30

Session:
Discrete PainlevÃ©

For each of the differential PainlevÃ© equations, the Riemann-Hilbert correspondence maps its solution space onto an affine cubic surface. In recent work with Nalini Joshi, a q-analog was established for the q-PainlevÃ© VI equation, where the associated algebraic surface is an affine Segre surface. In this talk, I will discuss this result and explain how the geometry of the Segre surface relates t
... More

Presented by Dr. Pieter ROFFELSEN
on
19 Jun 2023
at
16:30

Session:
Continuous integrable systems

Several examples of delay differential equations (equations relating a function of a single variable to shifts and derivatives of the function with respect to that variable) have appeared in the literature that deserve to be called integrable. Some are known to be reductions of integrable differential-difference equations and possess Lax pairs as well as continuum limits to the classical PainlevÃ
... More

Presented by Prof. Rod HALBURD
on
23 Jun 2023
at
15:30

Session:
Continuous integrable systems

The famous Peter-Weyl theorem tells that the ring of functions on a simple Lie group has a direct sum decomposition with summands isomorphic to tensor products of irreducible representations. In the case of more general groups there is no direct sum decomposition. However we prove that there exists a filtration such that its subquotients are tensor products of some very natural representations for
... More

Presented by Dr. Yevhen MAKEDONSKYI
on
23 Jun 2023
at
16:30

It is well known that the rational (or PadÃ©) approximants are closely related with the discrete-time Toda equation. The structural connection with orthogonal polynomials provides a link between the theory of integrable systems and various classical results of applied mathematics and numerical analysis. Inspired by recent advances on symmetry and integrability of difference equations I would like
... More

Presented by Prof. Adam DOLIWA
on
23 Jun 2023
at
09:45

Session:
Discrete PainlevÃ©

We consider two examples of certain recurrence relations, or nonlinear discrete dynamical systems, that appear in the theory of orthogonal polynomials, from the point of view of Sakaiâ€™s geometric theory of PainlevÃ© equations. Of particular interest is the fact that both recurrences are regularized on the same family of rational algebraic surfaces, but at the same time their dynamics are non-equ
... More

Presented by Dr. Anton DZHAMAY
on
19 Jun 2023
at
17:30

Session:
Continuous integrable systems

We establish all the reductions of a system of two coupled sine-Gordon equations introduced by Konopelchenko and Rogers to ordinary differential equations.
All these reductions are degeneracies of a master equation of Chazy,
"curious for its elegance", algebraic transform of the sixth PainlevÃ© equation.

Presented by Mr. Robert CONTE
on
23 Jun 2023
at
15:00

Session:
Ultra discrete equations

In the talk I will introduce a new version of the second main theorem in the tropical semiring and for tropical hypersurfaces. I will also present the inverse problem of tropical Nevanlinna theory. I shall show applications of tropical Nevanlinna theory for ultra-discrete PainlevÃ© equations.

Presented by Juho HALONEN
on
20 Jun 2023
at
15:20

Session:
Singularity analysis

We discuss singularity analysis and bilinear integrability of four
Bogoyavlensky differential-difference equations. Three of them are
completely integrable and the fourth is, to our knowledge, a new one.
Blending the singularity confinement with PainlevÃ© property reveals
strictly confining and anticonfining (weakly confining) singularity pat-
terns. The strictly confining patterns are useful
... More

Presented by Dr. Adrian Stefan CARSTEA
on
22 Jun 2023
at
10:15

Session:
Discrete differential geometry

In this talk we introduce a class of systems of difference equations defined on an elementary quadrilateral of the lattice and derive necessary conditions for their integrability. These conditions follow from the requirement that the system admits infinite hierarchies of symmetries, and can be used in the construction of the lowest order symmetries of the system. These considerations are demonstra
... More

Presented by Dr. Pavlos XENITIDIS
on
20 Jun 2023
at
16:30

A TCD map is a map from a triple crossing diagrams to projective space, satisfying an incidence requirement. We introduce dynamics on TCD maps based on Menelaus theorem and show multi-dimensionally consistency with Desargues theorem. To each TCD map we associate a hierarchy of dimer models, which provides local and global invariants. Conveniently, TCD maps include as special cases a large number
... More

Presented by Mr. Niklas AFFOLTER
on
23 Jun 2023
at
10:15

I will discuss a connection between the tetrahedron equation for maps
and the consistency property of integrable discrete equations on
$\mathbb{Z}^3$. The connection is based on the invariants of symmetry
groups of the lattice equations, generalizing a method developed in the
context of Yang-Baxter maps. The method will be demonstrated to certain
octahedron type lattice equations, leading to
... More

Presented by Dr. Anastasios TONGAS
on
19 Jun 2023
at
11:30

Session:
Compatibility

There is Hesse's principle of transfer: the transfer of geometric assertions from one dimension into another, facilitated by the fact that the projective subgroup stabilising a normal curve is isomorphic in every dimension. When space is coordinatised via a normal curve, geometric assertions become SL2-invariant equations, as opposed to homogeneous equations if a simplex is used. Although examples
... More

Presented by Mr. James ATKINSON
on
19 Jun 2023
at
10:30

Session:
Singularity analysis

Although the notion of singularity confinement was first introduced for the discrete KdV (dKdV) equation, as of yet there is still no rigorous definition of the notion of `confinement' for lattice equations. In fact, somewhat ironically, it has taken nearly 30 years before an exhaustive study of the singularities of the dKdV equation finally revealed their intriguing properties and the full richne
... More

Presented by Prof. Ralph WILLOX
on
22 Jun 2023
at
09:15

Session:
Non-linear waves

Modulation instability and nonlinearity are the main causes of the appearance of anomalous (rogue) waves (AWs) in several physical contexts. The theory of periodic anomalous waves has been recently developed on the basic Nonlinear SchrÃ¶dinger (NLS) model in 1+1 dimensions, adapting the finite gap method to the Cauchy problem for periodic initial perturbations of the homogeneous background solutio
... More

Presented by Prof. Paolo SANTINI
on
21 Jun 2023
at
09:15

The talk will discuss the emergence of factorised solutions of the Yang-Baxter equation in terms of transposition operators acting in q-difference (differential) representations of the algebra Uq(sln) (U(sln)), respectively. I will focus on the q-deformed case, where all but one of the transposition operators are constructed explicitly and the proof of a crucial symmetric group relation introduces
... More

Presented by Mr. Benjamin MORRIS
on
19 Jun 2023
at
12:30

Session:
Lagrangian multiform theory

The talk contains two parts. In the first part I will describe a bilinear framework for elliptic soliton solutions (which are composed by the LamÃ©-type plane wave factors and expressed using Weierstrass functions). The framework includes tau functions in Hirotaâ€™s form, vertex operators to generate such tau functions and the associated bilinear identities. These are introduced in detail for the
... More

Presented by Prof. Da-jun ZHANG
on
21 Jun 2023
at
12:30