Integrable Systems 2015

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We hosted our third annual Integrable Systems Workshop on 3 – 4 December 2015, at the School of Mathematics and Statistics, the University of Sydney.
Tropic curves on Confocal Quadrics in the Minkowski Space



Vladimir Bazhanov: Yang-Baxter maps, discrete integrable equations and quantum groups
For every quantized Lie algebra there exists a map from the tensor square of the algebra to itself, which by construction satisfies the set-theoretic Yang-Baxter equation. This map allows one to define an integrable discrete quantum evolution system on quadrilateral lattices, where local degrees of freedom (dynamical variables) take values in a tensor power of the quantized Lie algebra. The corresponding equations of motion admit the zero curvature representation. The commuting Integrals of Motion are defined in the standard way via the Quantum Inverse Problem Method, utilizing Baxter’s famous commuting transfer matrix approach. All elements of the above construction have a meaningful quasi-classical limit. As a result one obtains an integrable discrete Hamiltonian evolution system, where the local equation of motion are determined by a classical Yang-Baxter map and the action functional is determined by the quasi-classical asymptotics of the universal R-matrix of the underlying quantum algebra. In this paper we present detailed considerations of the above scheme on the example of the algebra Uq(sl(2)) leading to discrete Liouville equations, however the approach is rather general and can be applied to any quantized Lie algebra.
Raphael Boll: Two-dimensional variational systems on the root lattice Q(AN)
We study certain two-dimensional variational systems, namely pluri-Lagrangian systems on the root lattice Q(AN). Here, we follow the scheme which was already used to define two-dimensional pluri-Lagrangian systems on the cubic lattice ZN and three-dimensional pluri-Lagrangian systems on the lattice ZN as well as on Q(AN). We will show that the two-dimensional pluri-Lagragian systems on Q(AN) are more general, in the sense that such a system can encode several different pluri-Lagrangian systems on ZN. This also means that the variational formulation of several systems of certain hyperbolic equations, so-called quad-equations, can be obtained from one and the same pluri-Lagrangian system on Q(AN).
Peter Forrester: Random matrix theory and the Riemann zeros
Odlyzko has computed a data set listing more than 109 successive Riemann zeros, starting at a zero number beyond 1023. The data set relates to random matrix theory since, according to the Montgomery-Odlyzko law, the statistical properties of the large Riemann zeros agree with the statistical properties of the eigenvalues of large random Hermitian matrices. Moreover, Keating and Snaith, and then Bogomolny and collaborators, have used N×N random unitary matrices to analyse deviations from this law. Two contributions to this line of study are made. First, we point out that a natural process to apply to the data set is to thin it by deleting each member independently with some specified probability, and we proceed to compute empirical two-point correlation functions and nearest neighbour spacings in this setting. Second, we show how to characterise the order 1/N2 correction term to the spacing distribution for random unitary matrices in terms of a second order differential equation with coefficients that are Painlevé transcendents, and where the thinning parameter appears only in the boundary condition. This equation can be solved numerically using a power series method. Comparison with the Riemann zero data shows accurate agreement.
Claire Gilson: Generalizing Dodgson Condensation to Pfaffians and Quasideterminants
Charles Dodgson who was more famously known as Lewis Carroll was in fact a mathematician. He worked on matrices and introduced the concept of determinantal condensation.
This was a simple method that enabled easy enumeration of a determinant containing numbers. It uses an iterative process based upon Jacobi’s bilinear identity. In this talk I will introduce the concept of Dodgson condensation and look at some generalisations to pfaffians and quasideterminants which will be based on pfaffian and quasideterminant identities.
Giorgio Gubbiotti: The Generalized Symmetries of Boll equations
In [R. I.Yamilov, Uspekhi Mat. Nauk. (1983)] were classified all the differential difference equations of the class un˙= f(un-1,un,un+1) using the generalized symmetry method. The main result of this work up to Miura transformation, is the Toda equation and the so called Yamilov discretization of the Krichever Novikov equation (YdKN). In [D. Levi and R. I. Yamilov, J. Math. Phys. (1997)] a non-autonomous generalization of the YdKN equation was found.
The YdKN equation is of great importance in the theory of partial difference equations, since it was proved [D. Levi, M. Petrera, C. Scimiterna and R. I. Yamilov, SIGMA (2008)] that particular cases of it appear as three-point generalized symmetry of the equations belonging to the ABS classification [V. E. Adler, A. I. Bobenko and Y. B. Suris. Comm. Math. Phys. (2003)]. We show how the non-autonomous version of the YdKN equation appears in the generalized symmetries of the independent equations belonging to the classification of equations Consistent Around the Cube made my Raphael Boll in its PhD dissertation (2012) expanding the work of [P. D. Xenitidis and V. G. Papageorgiou, J. Phys. A: Math. Theor. (2009)] where was treated a single class of these equations.
Khaled Hamad and Peter van der Kamp: Laurentifying the symmetric QRT-map
Ultra-discretising difference equations yields bounds on degrees and on multiplicities of divisors. Recursive factorisation can then be used to prove polynomial upper bounds on growth of degrees. Moreover, it enables one to obtain recurrence equations for the divisors of the homogenised equation. When the difference equation is confining the order of the obtained recursion is fixed. Such recurrences are likely Laurent.
Chris Lustri: Stokes Switching in Discrete Painlevé Equations
Determining the asymptotic behaviour of solutions to the classical continuous Painlevé equations, such as the tronquée and tritronquée solutions to the first Painlevé equation, is a well-studied problem in integrable systems. However, there are few equivalent results currently available for the discrete Painlevé equations.
In this talk, I will discuss the use of exponential asymptotic methods for obtaining the asymptotic behaviour of solutions of discrete equations, and uncovering the Stokes Phenomena effects that are present in these solutions. From a careful analysis of the switching behaviour that occurs across Stokes lines, I am able to determine the regions of validity for the asymptotic solutions. I will demonstrate these techniques on several examples of discrete Painlevé equations.
Andrzej J. Maciejewski: Analytical method for spectra calculations in the Bargmann representation
We formulate a general method for determination of the closed form expresions for spectra of a class of quantum optics systems such as one- and multi-photon Rabi models, or N level systems interacting with a single mode of the electromagnetic field and their various generalisations, given in the Bargmann-Fock representation. In this representation the eigenproblem of the considered quantum models is described by a system of linear differential equations with one independent variable. The system can possess an irregular singular point at infinity. We explain three types of conditions that determine the spectrum: local and global conditions ensuring that the wave function is entire and normalization conditions stating that the entire function obtained in the above way must have a finite Bargmann norm. In the case of conditions that guarantee that wave function is entire Stokes phenomena are involved. Examples of specta determination by means of these conditions will be presented.
Ian Marquette: On superintegrable systems with a fourth order integrals of motion and connected to Painlevé transcendents
It has been discovered in 2004 by gravel that two-dimensional superintegrable systems allowing a second and third order integrals of motion on two-dimensional Euclidean space and allowing separation of variables in Cartesian coordinates involved the first, second and fourth Painlevé transcendent. All these systems can be shown to be reducible, and can be constructed from properties of 1D quantum systems. I will discuss how these ideas can be extended to the search and classification of systems with a fourth order integrals of motion. I will also discuss recent results using a direct approach to the search of such systems.
Joint work with Pavel Winternitz and Masoumeh Sajedi from Université de Montreal.
Davide Masoero: Bethe Ansatz and the Spectral Theory of affine Lie algebra-valued connections
To any simple Lie algebra we associate a connection with values in an affine Lie algebra. We prove that subdominant solutions to the ODE defined in different fundamental representations satisfy a set of quadratic equations called $\Psi$-system. This allows us to show that the generalized spectral determinants satisfy the Bethe Ansatz equations. Langlands duality is the key tool to deal with not-simply-laced Lie algebras.
Joint work with A. Raimondo and D. Valeri.
Akane Nakamura: Autonomous limit of the 4-dimensional Painlevé-type equations and degeneration of curves of genus two
The Painlevé equations have been generalized from various aspects. Recently, the 4-dimensional Painlevé-type equations were classified by corresponding linear equations (Sakai, Kawakami-N.-Sakai, Kawakami). In this talk, I explain an attempt to characterize the 40 types of integrable systems obtained as the autonomous limit of the 4-dimensional Painlevé-type equations, by inspecting the degenerations of their spectral curves, which are curves of genus two.
Hinke Osinga: Isochrons: computation and theory
We consider isochrons of a periodic orbit, which are manifolds of points that converge to the periodic orbit in phase with each other. We compute isochrons accurately and efficiently with a two-point boundary value problem set-up. We also define and compute isochrons of a focus equilibrium and isochrons of the reversed-time system, denoted backward-time isochrons. We show that a cubic tangency occurs between a set of forward-time and backward-time isochrons, which we call a cubic isochron foliation tangency (CIFT). This phenomenon is not a local feature but happens globally throughout the annulus where both sets of isochrons exist. As we show with examples of planar vector fields, the CIFT is the result of slow-fast time-scale dynamics.
Joint work with Bernd Krauskopf and Peter Langfield.
Maria Przybylska: Dynamics of a relativistic charge in classical Penning trap and Penning trap with inclined magnetic field
We study the dynamics of a classical charge within a processing chamber of two types of Penning traps: standard one with magnetic field exactly aligned with axis of symmetry of electrostatic quadruple potential and its generalization with magnetic field inclined under arbitrary angle to this axis. Relativistic Lagrangian and Hamiltonian dynamics without any approximations is analysed. For the standard Penning trap, thanks to the symmetry, reduction to a Hamiltonian system with two degrees of freedom is possible. The non-integrability of the reduced system is proved and some qualitative analysis of dynamics is presented. In the case when magnetic field is inclined also non-integrability is proved. Stability analysis shows the existence of critical inclination angle. If the inclination angle exceeds this critical value, the magnetron radius as well as the axial amplitude increase infinitely and a charge is lost from the trap. The resonant curves of the system are found. A comparison of the full non-linear dynamics with the averaged system is presented. In non-integrability proofs we use integrability obstructions due to properties of differential Galois groups of variational equations obtained from the linearisation along certain non-equilibrium particular solutions. Presented results were obtained in collaboration with Yurij Yaremko and Andrzej J. Maciejewski.
Reinout Quispel: Ruminations on a class of integrable Lotka-Volterra-like systems
Milena Radnovic: Elliptical billiards with Hooke's potential
We will review some geometrical and topological properties of such billiards.
Kouichi Takemura: New Airy-type solutions of the ultradiscrete Painlevé II equation with parity variables
The q-difference Painlevé II equation admits special solutions written in terms of determinant whose entries are the general solution of the q-Airy equation. An ultradiscrete limit of the special solutions is studied by the procedure of ultradiscretization with parity varialbes. Then we obtain new Airy-type solutions of the ultradiscrete Painlevé II equation with parity variables, and the solutions have richer structure than the known solutions. This talk is based on arXiv:1509.05248 jointly with Hikaru Igarashi, Shin Isojima.
Claude Viallet: Chasing singularities of rational discrete dynamical systems
Singularities play a fundamental role in the analysis of non linear differential and difference equations. For second order difference equations, or maps in two dimensions, the singularities have been studied extensively, using in particular blow up transformations and the theory of intersection of curves to characterise the singularity structure. This (almost) lead to a classification of the integrable cases, via the link to affine Lie algebras. The situation changes in higher dimension. I will show on the simple example of an integrable map in three dimensions that not all singular points play the same role. This is observed without performing explicit blow-up calculations. I will then describe what happens for a differential-difference equation, aka “delay Painlevé” equation, viewed as a map on infinite dimensional space.


Thursday, 3 December
Chair: Nalini Joshi
09:00 – 09:15 Opening: Joshi
09:15 – 09:45 Gilson
09:50 – 10:20 Viallet
10:25 – 10:55 Bazhanov
Morning tea
Chair: Claude Viallet
11:30 – 12:00 Quispel
12:05 – 12:35 Hamad & Van der Kamp
Chair: Nobutaka Nakazono
14:00 – 14:30 Takemura
14:35 – 15:05 Nakamura
15:10 – 15:40 Lustri
Afternoon tea
Chair: Holger Dullin
16:15 – 16:45 Maciejewski
16:50 – 17:20 Przybylska
Friday, 4 December
Chair: Reinout Quispel
09:00 – 09:30 Forrester
09:35 – 10:05 Osinga
10:10 – 10:40 Marquette
Morning tea
Chair: Claire Gilson
11:15 – 11:45 Masoero
11:50 – 12:20 Radnovic
Chair: Yang Shi
14:00 – 14:30 Boll
14:35 – 15:05 Gubbiotti
Afternoon tea and discussion


This workshop is being organised by Nalini JoshiChristopher Lustri and Milena Radnovic.
Second invariant curve for autonomous q-PIII in complex projective space


Register by emailing the organisers at:
Registrations closed on 1 November 2015.