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

Murray Batchelor: Some correlations for free parafermions

The calculation of various spin correlations for the Ising model uncovered deep connections with mathematics, notably to the theory of Painlevé equations. What makes the calculation of spin correlations possible for the Ising model is the underlying structure of free fermions. In recent years it has become apparent that there is a more general exactly solved Z(N) model described by free parafermions. For N = 2 this model includes the Ising model as a special case. This talk considers what (if anything) can be said about spin correlations for the Z(N) free parafermion model.

Vladimir Bazhanov: On the Yang-Baxter Poisson algebra in non-ultralocal integrable systems

A common approach to the quantization of integrable models starts with the formal substitution of the Yang-Baxter Poisson algebra with its quantum version. However it is difficult to discern the presence of such an algebra for the so-called non-ultralocal models. The latter includes the class of non-linear sigma models which are most interesting from the point of view of applications. In this work, we investigate the emergence of the Yang-Baxter Poisson algebra in a non-ultralocal system which is related to integrable deformations of the Principal Chiral Field.

Joshua Capel: Recent results in the construction of multivariate Racah polynomials

Racah polynomials are an important family of discrete orthogonal polynomials which arise when studying the Racah W-coefficients (or 6j symbols) arising in the theory of angular momentum. They also naturally arise when investigating the theory of second-order superintegrable systems.

Some multivariate versions of the Racah polynomials have been known about since the 90’s, appearing in the work of M. V. Tratnik. These multivariate version are conveniently obtained from expansions coefficients between particular bases in tensor product representations of su(2)/su(1,1). The standard method is to rewrite these expansion coefficients as a so-called vacuum coefficient multiplied by a polynomial. However there are many cases where the standard method fails to give polynomials.

This talk will present some recent results which examine why the standard method fails to give polynomials, and what modification to the standard method can be done to fix it (no specialist knowledge is assumed).

This is joint work with Sarah Post (University of Hawai`i at Mānoa).

Dmitry Demskoy: On non-steady motions of ideal fibre-reinforced fluids

We show that the system describing planar non-steady motions of fibre-reinforced fluids can be reduced, under certain conditions, to a single third order PDE in 1+1 dimensions. Remarkably, this equation has the same form as the equation that governs steady motions of fibre-reinforced fluids. Using this connection we propose a method of turning steady solutions into non-steady ones. The method is demonstrated by using various solutions of previously discovered integrable case of the equation. Elliptic and solitonic solutions are among them.

Yassir Dinar: Argument shift method and transverse Poisson structure of Lie-Poisson brackets

Let e be a nilpotent element of type D_{2n}(a_{n−1}) in a simple Lie algebra of type D_{2n}. Let Q be the Slodowy slice which is a transverse subspace to the adjoin orbit of e. We fix on Q compatible Poisson structures B_{1}^{Q} and B_{2}^{Q} where B_{1}^{Q} is the transverse Poisson structure of the Lie-Poisson bracket and B_{2}^{Q} depends on the properties of e. Using the argument shift method in this case, we obtain Liouville completely integrable systems P = (P_{1} , …, P_{3n−1} ) for B_{1}^{Q}. Our argument show that the sets of critical points of the corresponding momentum map P : Q → **C**^{3n−1} and the restriction of the adjoint quotient map χ : Q → **C**^{2n} coincide which implies that analyzing singularities of the two maps are equivalent. Generalize these results to other nilpotent elements is under investigation.

Jan de Gier: GUE distribution in a two-species exclusion processes

We compute the exact Green’s function as well as a joint current probability distribution of an exclusion process with two families of particles. We derive the asymptotic hydrodynamic current distribution function which is given by a mix of a Gaussian and a Tracy-Widom GUE distribution.

Giorgio Gubbiotti: Integrable discrete autonomous quad-equations admitting, as generalized symmetries, known five-point differential-difference equations

In this talk we present the construction of a family of autonomous quad-equations which admit as symmetries the five-point differential-difference equations belonging to known lists found by Garifullin, Yamilov and Levi. The obtained equations are classified up to autonomous point transformations and some simple non-autonomous transformations. We discuss our results in the framework of the known literature. There are among them a few new examples of both sine-Gordon and Liouville type equations.

Joint work with R. N. Garifullin and R. I. Yamilov.

Andrew Hone: Peakon solutions in integrable and non-integrable systems

Peakons are peaked solitons with a discontinuous first derivative at the peaks. They were first discovered by Camassa and Holm almost thirty years ago, arising as weak solutions in the dispersionless limit of an integrable partial differential equation (PDE) derived from shallow water theory. In this talk we review the properties of the Camassa-Holm and some analogous integrable PDEs. We further discuss how peakon sectors also appear as stable solutions in various non-integrable systems, such as the b-family of shallow water equations, and a coupled system due to Popowicz.

Kenji Kajiwara: Extensions of log-aesthetic curves in industrial design by integrable geometry

In this talk we discuss some extensions of the log-aesthetic curves (LAC) in industrial design based on the integrable deformation theory of plane curves under the similarity geometry. In this framework, LAC is understood as an analogue of the Euler’s elasticae in the sense that it allows two basic characterizations: (i) the rigid motion of integrable deformation of plane curves (ii) variational principle. Then we discuss two new extensions: (i) integrable discretization and (ii) space curve extension. In particular, the space curve version is characterized by the traveling wave solution of the coupled system of the mKdV and the third order Burgers equation, which is given in terms of the elliptic function and the Lamé function.

Peter van der Kamp: Reduction in the presence of integrable boundary

Integrable boundary consistency was introduced by Caudrelier, Crampé and Zhang in the paper [Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency, Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10, 2014]. In this talk, we present a new hierarchy of reductions, i.e. maps which represent reflection between integrable boundaries.

Chris Lustri: Generalised solitary waves in discrete integrable systems

Generalized solitary waves are nonlinear travelling wave with a central core that exhibits classical solitary wave behaviour, as well as non-decaying oscillations that continue away from the core indefinitely in one or both directions, and have amplitude that is exponentially small in some asymptotic parameter. This behaviour is typically associated with singular perturbations of some system, where the unperturbed system has classical solitary wave solutions.

I will show that discretizations of the KdV equation produce generalized solitary wave solutions in the continuum limit, even if the equation being discretized is not singularly perturbed, nor does it have some natural small parameter. I will then discuss the effect of discretization on relevant bifurcation parameters, by comparing the behaviour of the continuous and discretized versions of a singularly-perturbed fifth-order KdV equations.

I will conclude that exponentially-small oscillations are often caused by the discretization itself, and that it is possible to find special discretizations which either minimize or cancel these oscillations entirely. If time permits, I will further demonstrate this by considering an example advance-delay version of the saturated NLS equation, and showing that particular choices of the discretization parameter minimise the oscillations present in the solution.

This talk will discuss work performed in conjunction with N. Joshi, as well as work performed with G. L. Alfimov, A. S. Korobeinikov, and D. E. Pelinovsky.

Vladimir Mangazeev: Boundary matrices for the six-vertex model

We will consider solutions to the reflection equation related to the higher spin 6-vertex model. We will derive coupled q-difference equations for matrix elements of boundary K-matrices for arbitrary spin and express the solution in terms of hypergeometric functions. The case of stochastic zero-range processes will be also briefly discussed.

Reinout Quispel: Finding rational integrals of rational discrete maps

We present a novel method for finding rational first and second integrals of rational maps/ordinary difference equations.

Yang Shi: Certain subgroups of Coxeter groups and symmetry of integrable equations

The symmetries of many classes of integrable systems are characterised by Coxeter groups, also known as reflection groups, which have remarkable properties. Can we exploit these properties to help us to answer questions arose in the context of integrable systems? Here we described the Nomalizer theory of parabolic subgroups of the Coxeter groups and show how it can be used to establish explicite relations between different classes of equations found in the integrable system literature.

Andreas Vollmer: An algebraic-geometric approach to the classification of superintegrable systems

During the recent years the classification of superintegrable systems has undergone significant activity. Most importantly, superintegrable systems on conformally flat manifolds in dimension 2 and 3 have been classified, and surprising links to hypergeometric polynomials and other areas have been discovered. However, the current approach does not extend to higher dimensions, even if heavy computer algebra is employed.

In my talk I will discuss, for arbitrary dimension, how the classification problem can be reduced to a set of simple (algebraic) equations. This paves the way to a full classification of superintegrable systems in terms of an algebraic variety.

(this is a joint project with Jonathan Kress and Konrad Schoebel).

Michael Twiton: A dynamical systems approach to the fourth Painlevé equation

We use methods from dynamical systems to study the fourth Painlevé equation P_{IV}. Our starting point is the symmetric form of P_{IV}, to which the Poincaré compactification is applied. The motion on the sphere at infinity can be completely characterized. There are 14 equilibrium points, which are classified into three different types. Generic orbits of the full system are curves from 1 of 4 asymptotically unstable points to 1 of 4 asymptotically stable points, with the set of transitions that are allowed depending on the values of the parameters. This allows us to give a qualitative description of a generic real solution of P_{IV}.

Joint work with Jeremy Schiff.

Da-jun Zhang: On rational solutions of discrete integrable systems

Hirota-Miwa equation (also known as discrete AKP equation) is one of general 3D discrete integrable equations. Tau function of this equation admits an algebraic form, composed by polynomials of discrete independent coordinates. In this talk I will discuss properties of such a tau function and its applications in constructing rational solutions of integrable quadrilateral equations (such as the Nijhoff-Quispel-Capel equation, equations in the Adler-Bobenko-Suris (ABS) list and some multi-quadratic ABS equations). These results can be extended to nonautonomous case.

The Workshop will be held in New Law School Annexe, Level 3, Seminar room 342.

Registrations close on 1 November 2018.