We are hosting our sixth annual Workshop on Integrable Systems on

**29 – 30 November 2018**

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

#### Organisers

This workshop is being organised by Giorgio Gubbiotti, Nalini Joshi, Milena Radnovic, Yang Shi, Stephanie Swanson, and Dinh Tran.

#### Participants

Murray Batchelor, Australian National University
Vladimir Bazhanov, Australian National University
Amin Chabchoub, University of Sydney
Dmitry Demskoy, Charles Sturt University
Yassir Dinar, Sultan Qaboos University

Jan de Gier, University of Melbourne
Giorgio Gubbiotti, University of Sydney
Zeina Haidar, University of Sydney
Khaled Hamad, La Trobe University

Andrew Hone, University of Kent
Nalini Joshi, University of Sydney
Kenji Kajiwara, Kyushu University
Chris Lustri, Macquarie University
Vladimir Mangazeev, Australian National University
Ian Nagle, Macquarie University

Reinout Quispel, La Trobe University
Milena Radnovic, University of Sydney
John Roberts, UNSW
Yang Shi, University of Sydney
Dinh Tran, University of Sydney
Michael Twiton, Bar-Ilan University

Andreas Vollmer, UNSW

Asem Wardak, University of Sydney
Dongping Yang, University of Sydney
Da-jun Zhang, Shanghai University
#### Talks

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.

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.

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.

John Roberts: Using Ideal Theory to study Discrete Integrable Systems

Looking for rational integrals of rational maps is a natural exercise that benefits from applying results from ideal theory, as I’ll describe.

Michael Twiton: A Dynamical Systems Approach to the Fourth Painlevé Equation

We use methods from dynamical systems to study the fourth Painlevé equation PⅣ. Our starting point is the symmetric form of PⅣ, 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 PⅣ.
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.

#### Registration

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

Registrations close on 1 November 2018.