Integrable Systems 2016

Organisers | Participants
Schedule | Talks | Special lecture series
Previous workshops | Registration

We hosted our fourth annual Workshop on Integrable Systems on

1 – 2 December 2016

at the School of Mathematics and Statistics, the University of Sydney.

5-cube

Organisers

This workshop is being organised by Nalini JoshiChristopher Lustri, Nobutaka NakazonoMilena Radnovic, Yang Shi, and Stephanie Swanson.

Exponentially small waves present in the Frenkel Kontorova Model

Participants

Schedule

Thursday, 1 December
Chair: Nalini Joshi
09:00 – 09:15 Opening: Joshi
09:15 – 09:45 Guttmann
09:50 – 10:20 Bazhanov
10:25 – 10:55 Braden

Morning tea

Chair: John Roberts
11:30 – 12:00 Siu
12:05 – 12:35 Dullin

Lunch

Chair: Yasuhiko Yamada
14:00 – 14:30 Batchelor
14:35 – 15:05 Mangazeev
15:10 – 15:40 Bosnjak

Afternoon tea

Chair: Da-jun Zhang
16:15 – 16:45 Warnaar
16:50 – 17:20 de Gier

Friday, 2 December
Chair: Ian Marquette
09:00 – 09:30 Yamada
09:35 – 10:05 Sekiguchi
10:10 – 10:40 Suzuki

Morning tea

Chair: Nobutaka Nakazono
11:15 – 11:45 Zhang
11:50 – 12:20 Shi

Lunch

Chair: Omar Foda
14:00 – 14:30 Demskoy
14:35 – 15:05 Hlavac

Afternoon tea

Chair: Vladimir Bazhanov
15:40 – 16:10 Runov
16:15 – 16:45 Foda

The Workshop will be held in New Law School Annexe, Level 4, Seminar room 444.

Configuration of the bilinear discrete KP equation

Talks

Murray Batchelor: Free parafermions
I will discuss some properties of a remarkably simple N-state spin chain discovered by Rodney Baxter in 1989 which Paul Fendley pointed out recently is described by free parafermions.
Vladimir Bazhanov: Bukhvostov-Lipatov model and duality of quantum and classical systems
Bukhvostov and Lipatov have shown that weakly interacting instantons and anti-instantons in the O(3) non-linear sigma model in two dimensions are described by an exactly soluble model containing two coupled Dirac fermions. We propose an exact formula for the vacuum energy of the model for twisted boundary conditions, expressing it through a special solution of the classical sinh-Gordon equation. The formula perfectly matches predictions of the standard renormalized perturbation theory at weak couplings as well as the conformal perturbation theory at short distances. The description of the vacuum state energy of the quantum model in terms of the classical sinh-Gordon equation can be viewed as an instance of a remarkable, albeit unusual correspondence between integrable quantum field theories and integrable classical field theories in two dimensions, which cannot be expected from the standard quantum–classical correspondence principle.
Gary Bosnjak: Construction of R-Matrices related to sl(n)
In this talk we will overview a 3D approach to constructing R-matrices related to Uq(sln). We derive an explicit formula for matrix elements in the case of symmetric tensor representations with arbitrary weights. We will show this formula has a neat factorised form which allows us to give a simple proof that the R-Matrix can be stochastic. We may also mention some alternative constructions and talk about generalising these results.
Harry Braden: On the Construction of Monopoles
Although the study of BPS monopoles is now over 30 years old there are still few analytic results known for the Higgs and gauge fields. For su(2) monopoles without spherical or axial symmetry the only known results are for the Higgs field on a coordinate axis for charge 2. By combining integrable systems and twistor constructions we show how the problem becomes algebraic in an appropriate gauge. The approach will be illustrated by presenting the general charge 2 fields.
Dmitry Demskoy: Recursion operators of hyperbolic equations
A recursion operator encodes information about the whole hierarchy of integrable equations. In principle, it allows one to study an equation with the same fullness as it is possible by using a Lax pair. We observe that recursion operators of hyperbolic equations that degenerate into Darboux-integrable equations have a special structure. This observation is used to find new recursion operators for some Lagrangian systems of sine-Gordon type.
Holger Dullin: Toric, Semi-Toric, and beyond
Toric integrable systems with their momentum maps whose images are rational polytopes are well understood. More recently San Vu Ngoc and Alvaro Pelayo classified semi-toric system in two degrees of freedom with additional singularities of focus-focus type. I will briefly review both toric and semi-toric systems. Then I will show that semi-toric systems can always be deformed such that the global circle action remains intact, but the focus-focus point is replaced by an elliptic-elliptic point and possibly additional singularities, some of which are hyperbolic (joint work with Alvaro Pelayo [1]). This deformation is inspired by the Hamiltonian Hopf bifurcation, well known in dynamical systems. This mechanics can be used to deform toric systems into semi-toric systems, and also to deform semi-toric systems into hyperbolic semi-toric systems. I will discuss some examples of such deformations (joint work with Joachim Worthington [2]) and the behaviour of the joint spectrum of the corresponding integrable quantum systems.

[1] Holger R. Dullin and Alvaro Pelayo. Generating hyperbolic singularities in semitoric systems via Hopf bifurcations. Journal of Nonlinear Science, 26(3), pp 787-811, 2016.
[2] Holger R. Dullin and Joachim Worthington. The polygonal invariant of a deformed spin-oscillator with hyperbolic singularities. (in preparation)
Omar Foda: Refining the topological vertex
I would like to introduce the topological vertex, explain how it can be used to construct conformal blocks in 2D conformal field theories, recall its known refinement, then discuss an additional Macdonald-type refinement recently obtained in joint work with J F Wu (Beijing).
Jan de Gier: Matrix product for Macdonald polynomials and generalisations
New formulas for Macdonald polynomials were recently found using matrix products based on high rank bosonic L-matrix solutions of the Yang-Baxter equation. These formulas lead to natural generalisations of Macdonald polynomials using other L-matrix solutions, and in particular unify Macdonald polynomials with recently defined spin s generalisations of Hall-Littlewood polynomials by Borodin and Petrov.
Anthony J. Guttmann: Pattern-avoiding permutations and integrability
Permutations that avoid sub-permutations have a long history in algebraic combinatorics and theoretical computer science. Some such problems are solvable and others appear not to be. While there is no natural Tang-Baxter equation, a number of problems in statistical mechanics and other fields can be related to pattern-avoiding permutations. We will discuss these connections, and give some results that we have recently obtained.
Joint work with Andrew Elvey-Price and Andrew Conway, all from The University of Melbourne.
Adam Hlavac: On the constant astigmatism equation and surfaces of constant astigmatism
Surfaces of constant astigmatism, i.e. surfaces characterized by the condition ρ2−ρ1=const≠0, where ρ1, ρ2 are the principal radii of curvature, were already known at the end of the nineteenth century. They appear in the context of pseudospherical surfaces, which, themselves, correspond to solutions of the sine-Gordon equation uξη=sin u.
In 2009, after a century in oblivion, the subject of constant astigmatism surfaces has been resurrected by H. Baran and M. Marvan in their work concerning the systematic search for integrable classes of Weingarten surfaces. Constant astigmatism surfaces correspond to solutions of the integrable equation zyy+(1/z)xx+2=0 called the constant astigmatism equation (CAE). Solutions of the CAE can be interpreted as spherical orthogonal equiareal patterns, with relevance to two-dimensional plasticity.
In the talk, an overview of new results published in [1, 2] will be given.
1. We introduce an algebraic formula producing infinitely many exact solutions of the CAE from a given seed. A construction of corresponding surfaces of constant astigmatism is then a matter of routine. As a special case, we consider multisoliton solutions of the constant astigmatism equation defined as counterparts of famous multisoliton solutions of the sine-Gordon equation.
2. For the CAE, we construct a system of nonlocal conservation laws (an abelian covering) closed under the reciprocal transformations. We give functionally independent potentials modulo a Wronskian type relation.
[1] A. Hlavac, On multisoliton solutions of the constant astigmatism equation, J. Phys. A: Math. Theor. 48 (2015) 365202.
[2] A. Hlavac and M. Marvan, Nonlocal conservation laws of the constant astigmatism equation, arXiv:1602.06861 (2016)
Vladimir Mangazeev: Stochastic R matrix for Uq(An(1))
We show that the quantum R matrix for symmetric tensor representations of Uq(An(1)) satisfies the sum rule required for its stochastic interpretation under a suitable gauge. Its matrix elements at a special point of the spectral parameter are found to factorize into the form that naturally extends Povolotsky’s local transition rate in the q-Hahn process for n=1. Based on these results we formulate new discrete and continuous time integrable Markov processes on a one-dimensional chain in terms of n species of particles obeying asymmetric stochastic dynamics.
This a joint work with A. Kuniba, S. Maruyama and M. Masato.
Boris Runov: On spectral problem of integrable Quantum Field Theory
We study ODE/IM correspondence in QFT on the example of Lipatov-Boukhvostov model. We associate with QFT a second order differential equation with 3 singular points arising from linear problem for modified sinh-Gordon equation. Functional equations on connection matrices manifesting the symmetry of ODE w.r.t. rotation around singularities give rise to Bethe Ansatz type equations which are interpreted as Bethe Ansatz equations for Lipatov-Boukhvostov model. Using analytical properties of connection matrices we derive a system of nonlinear integral equations, which can be solved numerically. We compute vacuum energy of the model and check our results against asymptotical expressions obtained using perturbation theory.
Jiro Sekiguchi: Flat Structure and Potential Vector Fields related with Algebraic Solutions to Painleve VI Equation
A potential vector field is a solution of an extended WDVV equation which is a generalization of a WDVV equation. It would be expected that potential vector fields corresponding to algebraic solutions of Painleve VI can be written by using polynomials or algebraic functions explicitly. The purpose of this talk is to explain an idea to construct potential vector fields from free divisors show potential vector fields corresponding to some of non-equivalent algebraic solutions.
This is a joint work with M. Kato and T. Mano.
Yang Shi: Reflection groups and discrete integrable systems
Discrete integrable systems include many types of well-known nonlinear partial/ordinary difference equations such as: Hirota’s octahedron equation, the cross-ratio equation, and discrete analogues of the Painleve equations. During the last decade active investigations into the connections between the different classes of discrete equations have led to exciting discoveries. In this talk we discuss some recent progress concerning the relation between two classes of discrete integrable systems: a list of quadrilateral equations (known as the ABS equations) and Sakai’s classification of 22 types of discrete Painleve equations. This connection was made possible by exploiting a combinatorial/geometrical realisation of the Weyl groups.
Tim Siu: Detecting integrals in reversible maps
In this talk, we study discrete dynamical systems that are reversible mappings. By reducing to the finite field, we obtain a partition of the space into its constituent orbits. We will see that there is a strong relationship between the number of asymmetric periodic orbits and the number of algebraic integrals that the mapping has. By using this number, we develop a test to extract the number of integrals in reversible mappings.
Takao Suzuki: From Heine to q-Painleve
The q-Painleve VI equation was introduced by Jimbo and Sakai. It is known that it admits a particular solution in terms of the Heine’s basic hypergeometric function 2φ1. In this talk, we propose a higher order generalization of the q-Painleve VI equation which admits a particular solution in terms of n+1φn.
Ole Warnaar: The Nekrasov-Okounkov identity
The Nekrasov-Okounkov identity first arose in supersymmetric gauge theory as the partition function of a certain ensemble of random three-dimensional partitions. In this talk I will try to explain the Nekrasov-Okounkov identity from an integrable systems point of view, and discuss several generalisations, including a mysterious modular analogue.
Yasuhiko Yamada: On the q-Garnier system
The q-Garnier system [Sakai, 2005] is a q-difference and a multi-variable extension of the Painleve equations. In this talk I will discuss the q-Garnier system from the following points of view.(1) A simple form of the evolution equations and the Lax pair. (2) The autonomous limit: a generalization of the QRT system to hyper-elliptic curves. (3) Some special solutions obtained by Pade method.
This talk is based on the joint work with H.Nagao [arXiv 1601.01099].
Da-jun Zhang: On 3D consistency and 1-soliton solutions
In this talk I will review deriving seed solutions (0SS) and 1-soliton solutions (1SS) via the Backlund transformations (BT) and 3D consistency. It is known that (auto) BT and Miura transformation (non-auto BT) can be constructed for 2D lattice equations with 3D consistency. Using “fixed point” idea of BT one can derive 0SS and then from BT to derive 1SS. As an example we can look at Q1(δ) in the ABS list which can have two types of 0SSs and then two 1SSs [1]. In the talk we will construct a series of rational functions as new seed solutions of Q1 equation. Then from these solutions we get 1SSs for Q1, H3 and Q2 equations [3]. We will also look at elliptic 1SS of the dpKdV equation [2] and discrete Boussinesq (DBSQ) equation [4]. In this case we try to introduce elliptic dispersion relation for discrete Integrable systems. It is conjectured that for elliptic 1SSs, two spacing parameters in the lattice equation are covariant. Note that for ordinary 1SS these parameters are independent.
Main references:
1. J. Hietarinta, D.J. Zhang, Soliton solutions for ABS lattice equations: II: Casoratians and bilinearization, J. Phys. A: Math. Theor., 42(40), (2009) No.404006 (30pp).
2. F.W. Nijhoff, J. Atkinson, Elliptic N-soliton solutions of ABS lattice equations, Int. Math. Res. Not., (2010) No.10, 3837-95.
3. D.D. Zhang, D.J. Zhang, preprint, 2016.
4. F.W. Nijhoff, Y.Y. Sun, D.J. Zhang, preprint, 2016.

Blow up

Special lecture series

In the days preceding the Workshop, Yasuhiko Yamada will give a series of lectures on the Pade approach for constructing isomonodromic equations for nonlinear differential and discrete integrable systems. The lectures should be suitable for staff, post-graduate students and honours students.

Lecture series: Introduction to Pade method
Isomonodromic equations such as Painlevé and Garnier equations are very important class of nonlinear differential equations. On the discrete analog of these equations, much progress has been made over the last decades. In this series of lectures, I will explain a very simple method to approach the isomonodromy equations, both differential and discrete, based on the Padé approximation.
Monday, 28 November, 10am-11:30am: Pade approximation
For a given function ψ(x), the Padé approximation supply a rational function P(x)/Q(x) as an approximation of ψ(x). We consider the linear differential equations for y(x) satisfied by y(x)=P(x) and y(x)=ψ(x)Q(x), and explain how to compute them in explicit examples. By choosing the function ψ(x) suitably, the linear differential equations give the Lax pair for Painlevé type equations.
Monday, 28 November, 2pm-3:30pm: Schur function formula
In case of ψ(x)=Σpixi, an explicit formula of the polynomials P(x),Q(x) is known. The polynomials are given in terms of the some determinants (Schur functions) with entries pi. This formula is useful to obtain special solutions of the Painlevé type equations.
Tuesday, 29 November, 10am-11:30am: Discrete case
We consider the Padé approximation where the function ψ is given in terms of qq-Pochhammer symbols. Then the qq-difference Painlevé equations and their special solutions are obtained.
Tuesday, 29 November, 2pm-3:30pm: Padé interpolation
We study the discrete Painlevé equations by using the Padé interpolation. The Padé interpolation is a discrete version of the Padé approximation and it is older than the usual differential case. The explicit formula for P(x),Q(x) are known by Cauchy and Jacobi.
Wednesday, 30 November, 10am-noon: Generalizations
I will discuss various generalizations such as higher-order/higher-rank, qq-difference/elliptic-difference, multiple Padé approximation (by Hermite) etc, as long as time permits.
The special lecture series will be held in the New Law School Lecture Theatre 026.

Configuration of the bilinear discrete KP equation

Previous workshops

First Workshop on Integrable Systems, 2 December 2013
Second Workshop on Integrable Systems, 4-5 December 2014
Third Workshop on Integrable Systems, 3-4 December 2015

An approximation of a tronquée solution of the fourth Painlevé equation

Registration

Register by emailing the organisers at: integrable@maths.usyd.edu.au

Registrations close on 1 November 2016.

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