at the School of Mathematics and Statistics, the University of Sydney.
Nail Akhmediev: Nonlinear Schrodinger equation, its extensions and rogue wave solutions
It was found, recently, that certain types of extreme events can be described using solutions of the nonlinear Schrödinger equation (NLSE). There is a hierarchy of rational solutions of the focusing NLSE with increasing order and with progressively increasing amplitude. As the equation can be applied to waves in the deep ocean, the solutions can describe “rogue waves” with virtually infinite amplitude. They can appear from smooth initial conditions. Thus, a slight perturbation on the ocean surface can dramatically increase the amplitude of the singular wave event that appears as a result. The NLSE can be extended in order to describe the influence of higher-order effects on rogue waves. These extensions include Hirota equation, Sasa-Satsuma equation and their varieties. Moreover, infinite number of terms can be added to the NLSE keeping its main feature – integrability. All these equations have solutions in the form of rogue waves.
Vladimir Bazhanov: Towards Canonical Quantization of Non-Linear Sigma-Models
In this talk we revisit the problem of canonical quantization of two-dimensional non-linear sigma models (NLSM) in two dimensions. We unravel the integrable structure of the O(3) NLSM and its one-parameter deformation — the sausage model, by resolving the long-standing “non-ultralocality” problem. Our consideration is based on the continuous version of the the Quantum Inverse Scattering Method enhanced by a powerful ODE/IQFT correspondence, which connects stationary states of Integrable QFT models with special solutions of classical integrable equations. Among the obtained results is a system of non-linear integral equations for computation of vacuum eigenvalues of the continuous analogs of quantum transfer-matrices for the O(3)/sausage NLSM. The talk is based on the recent article arXiv:1706.09941 (joint with Gleb Kotousov and Sergei Lukyanov).
Emma Carberry: Toroidal Soap Bubbles: Constant Mean Curvature Tori in S3 and R3
Constant mean curvature (CMC) tori in S3, R3 or H3 are in bijective correspondence with spectral curve data, consisting of a hyperelliptic curve, a line bundle on this curve and some additional data, which in particular determines the relevant space form. This point of view is particularly relevant for considering moduli-space questions, such as the prevalence of tori amongst CMC planes and whether tori can be deformed. I will address these questions for the spherical and Euclidean cases, using Whitham deformations.
Amin Chabchoub: The significance of solitons and breathers in hydrodynamic applications
The dynamics of water waves can be described within the framework of weakly nonlinear evolution equations such as the Korteweg-de Vries equation (KdV) in shallow-water and the nonlinear Schrödinger equation (NLS) in intermediate water depth as well as deep-water regime. Both, the KdV and NLS are physically very rich and can be for instance used to study the fundamental principles of nonlinear dynamics such as the Fermi-Pasta-Ulam recurrence. By applying mathematical techniques such as the Darboux transformation or the inverse scattering transform, these integrable evolution equations provide exact models that can be studied analytically, numerically and controlled in laboratory environments. The motion of rogue waves has attracted the scientific interest lately. Indeed, one possible explanation their formation is provided by the modulation instability. This latter instability can be deterministically discussed within the context of exact NLS breather solutions, such as fundamental Akhmediev- or Peregrine-type breathers. Recent laboratory experiments on solitons and breathers in water wave facilities will be presented while new insights of the modulation instability will be discussed in detail.
Holger Dullin: Monodromy in the Kepler Problem
What could possibly be said about the Kepler Problem that is new?
It is well known that this superintegrable system can be separated in different coordinate systems, and each such separation defines a distinct Liouville integrable system. We show that for separation in prolate spheroidal coordinates the resulting Liouville integrable system has Hamiltonian monodromy, which means that the action variables cannot be globally smoothly defined. This has interesting consequences for the quantum mechanics of the problem, which we illustrate. Similar analysis can be done for many prominent superintegrable systems, for example the harmonic oscillator, the free particle or the geodesic flow on the sphere.
(joint work with Holger Waalkens, Groningen)
https://arxiv.org/abs/1612.00823 Charalambos Evripidou: Poisson structures for difference equations
In this work we study the existence of Poisson structures that are preserved by difference equations of a special form as well as the inverse problem, given a Poisson structure to find a difference equation preserving this structure. We give examples of quadratic Poisson structures that are preserved by maps of the same form as the KP maps which are obtained from a travelling-wave reduction of the corresponding integrable partial difference equation. We also give examples related to several well known maps such as the SG, KdV, mKdV and pKdV reductions.
Peter Forrester: Volumes and distributions for random lattices
Fundamental to random matrix theory are various factorisations of Lebesgue product measure implied by matrix change of variables. In number theory, factorisation of Siegel’s invariant measure for SLN(R) is an ingredient in Duke, Rudnik and Sarnak’s asymptotic computation of the number of matrices in SLN(Z), with a bounded norm. In this talk it will shown how factorisation of measure allows for calculations in the space of integral lattices SLN(R)/SLN(Z) and generalisations such as SLN(C)/ SLN(Z[i]).
Andrew Hone: Linearization and reduction of a 6-point lattice equation
We present a lattice equation on a 6-point stencil, whose reductions include a family of recurrences with the Laurent property that were found in the thesis of Ward, and also in a classification by Alman et al. of period 1 seeds in Laurent phenomenon (LP) algebras. It is shown that the lattice equation is linearizable, and the corresponding results for general travelling wave reductions and their properties are also presented. In particular, we find a mechanism for the Laurent property that lies outside the usual one for cluster algebras, or LP algebras.
Rei Inoue Yamazaki: Cellular automata for reduced words in the affine symmetric group
Recently Glick and Pylyavskyy introduced a new method to define integrable rational maps from pairs of reduced elements in the affine symmetric group, where the network on a cylinder plays an important rule. We discuss the `tropicalization’ of a specific family of the maps, and show that its soliton solutions are related to the combinatorial R-matrix acting on a tensor product of the affine sln-crystal of one-row Young tableaux. Especially, in a sense the soliton solutions are `dual’ to those for the box-ball system.
This talk is based on a joint work with Max Glick and Pavlo Pylyavskyy.
Nalini Joshi: Elliptic-difference-type Painlevé equations
At the head of the list of discrete Painlevé equations described by Sakai (2001) sits an elliptic difference equation, which has attracted a great deal of attention in recent times. Since its discovery, a perennial question has been whether it is unique as the only integrable elliptic-difference-type equation. I will report on results obtained with Atkinson, Howes, Nakazono and describe two new examples of elliptic-difference equations investigated in our recent papers. One of these is a new elliptic-difference equation.
Peter van der Kamp: Families of integrable mappings of the plane
I will present families of maps of the plane which:
-are a composition of two generalised Manin involutions
-preserve a pencil of polynomial curves of certain degree, and
-are measure preserving.
The QRT-family arise as a special case of the degree 4 case, in a limit where two singular base points go to infinity. If time permits I will also discuss
-subfamilies which admit a root (generalisations of symmetric QRT), and
-involutions where the involution point is not fixed.
The talk is based on joint work with D.I. McLaren and G.R.W. Quispel.
Spyridon Kamvissis: Initial/Boundary Value Problems in Soliton Theory: Admissible Data for the Unified Scattering Method
Initial/boundary value problems for 1-dimensional `completely integrable’ equations (NLS, KdV, Sine-Gordon, etc.) can be studied via an extension of the inverse scattering method, which is due to Fokas and his collaborators. A crucial feature of this method is that it requires the values of more boundary data than given for a well-posed problem. In the particular case of cubic NLS, for example, knowledge of the Dirichlet data suffices to make the problem well-posed but the Fokas “unified” method also requires knowledge of the values of Neumann data. The study of the Dirichlet to Neumann map is thus necessary before the application of the unified method. In recent work with Dimitra Antonopoulou, we provide a rigorous study of this map for a large class of decaying Dirichlet data. We show that the Neumann data are also sufficiently decaying and hence rigorously justify the applicability of the unified method. In a sense we rigorously prove the complete integrability of the Initial/Boundary Value Problem.
References:
Nonlinearity 28 (2015) 3073-3099
Nonlinearity 29 (2016) 3206-3214
Decio Levi: Classification of integrable differential-difference equations relating five lattice point
Using the generalized symmetry method we carry out a classification, of integrable autonomous five-point differential-difference equations. The resulting list, up to autonomous point transformations, contains 31 equations some of which seem to be new. We have found non-autonomous or non-point transformations relating most of the resulting equations among themselves as well as their generalized symmetries.
Oleg Lisovyi: Painlevé functions, Fredholm determinants and combinatorics
I am going to explain explicit construction of general solutions to isomonodromy equations, with the main focus on the Painlevé VI equation. I will start by deriving Fredholm determinant representation of the Painlevé VI tau function. The corresponding integral operator acts in the direct sum of two copies of L2(S1). Its kernel is expressed in terms of hypergeometric fundamental solutions of two auxiliary 3-point Fuchsian systems whose monodromy is determined by monodromy of the associated linear problem via a decomposition of ℂℙ1\{4 points} into two pairs of pants. In the Fourier basis, this kernel is given by an infinite Cauchy matrix. I am going to show that the principal minor expansion of the Fredholm determinant yields a combinatorial series representation for the general solution to Painlevé VI in the form of a sum over pairs of Young diagrams.
Jean-Marie Maillard: Modular forms, Schwarzian derivative conditions, and symmmetries of differential equations in physics
We first illustrate covariance properties on order-two linear differential operators associated with identities relating the same 2F1 hypergeometric function with different rational pullbacks. These rational transformations are solutions of a differentially algebraic equation that already emerged in a paper by Casale on the Galoisian envelopes. We then focus on identities relating the same 2F1 hypergeometric function with two different algebraic pullback transformations: such remarkable identities correspond to modular forms, the algebraic transformations being solution of another differentially algebraic Schwarzian equation that also emerged in Casale’s paper.
A generalization of such a simple covariance is seen to be the well-suited concept for representations of the renormalization group for integrable models.
Ian Marquette: Higher rank quadratic algebras and algebraic derivation of spectrum
Quadratic and more generally finitely generated polynomial algebras appear to be naturally related with integrals of superintegrable models. These models have been connected with the full Askey scheme of orthogonal polynomials, exceptional orthogonal polynomials, Painleve transcendents and equations classified by Chazy. I will review some of the recent year results on N-dimensional superintegrable systems. In particular a work on a higher rank quadratic algebra with an embedded structure.
Reinout Quispel: On a dual to the lattice AKP equation
We present a 3D lattice equation which is dual to the lattice AKP equation. Reductions of this equation include Rutishauser’s quotient-difference (QD) algorithm, the higher analogue of the discrete time Toda (HADT) equation and its corresponding quotient-quotient-difference (QQD) system, the discrete hungry Lotka-Volterra system, discrete hungry QD, as well as the hungry forms of HADT and QQD. We provide three conservation laws, we propose an N-soliton solution, and we conjecture that (periodic) reductions to ordinary difference equations have the Laurent property.
This is joint work with Peter van der Kamp and Da-jun Zhang.
Dinh Tran: Poisson brackets for periodic reductions of (integrable) lattice equations
In this talk, I will present Poisson brackets of mappings obtained as reductions of lattice equations. Poisson brackets for can be derived from the existence of Lagrangians.
Peter Vassiliou: Role of Symmetry in Nonlinear Control Theory
A control system is a dynamical system containing forcing terms that can be continuously varied to achieve desirable prescribed trajectories in the underlying dynamical process. A typical example is balancing a broomstick on your finger. Such systems are ubiquitous in many technologies as well as in daily life. This has led to a vast multidisciplinary enterprise with many facets and points of view.
An important question in nonlinear control theory that has attracted considerable interest is the notion of explicit Integrability. For instance one would like to know when a control system is explicitly integrable. In this talk I will explain the issues and show how recent work involving symmetry notions has shed light on the explicit integrability question. Time permitting I’ll attempt to remark on some of the many open problems.
Claude Viallet: From maps with invariants to invariants with maps
We provide new examples of rational maps in four dimensions with two rational invariants, which have unexpected geometric properties and fall outside classes studied by earlier authors. We can reconstruct the map from both invariants, but one of the invariants also defines a new map, which we call the shadow map. We show how to go beyond ellipticity, and address, once more, the fate of “singularity confinement”.
(see arXiv:1706.00173) Alessandra Vittorini Orgeas: Yang-Baxter solution of Dimers as a Six-Vertex Model
Using Yang-Baxter integrability we study dimers as a free-fermion six-vertex model with crossing parameter λ = π/2. A one-to-many mapping of vertex onto dimer configurations allows the free-fermion solutions to be applied to the anisotropic dimer model on a square lattice where the dimers are rotated by 45o compared to their usual orientation. The dimer model is described by a fermion algebra and the Temperley-Lieb algebra with loop fugacity β = 2 cos λ = 0. Using commuting transfer matrices, we establish and solve inversion identities on the torus for arbitrary finite size. By calculating the partition function at the isotropic point u = π/4, we obtain an explicit formula for the counting of the rotated dimer configurations on a finite M × N periodic lattice. Remarkably, the modular invariant partition function on the torus is the same as symplectic fermions and critical dense polymers. On the strip, with vacuum boundary conditions, the dimer Hamiltonian and double row transfer matrices exhibit nontrivial Jordan cells. We therefore argue that, in the continuum scaling limit, the dimer model gives rise to a logarithmic conformal field theory with central charge c = −2, minimal conformal weight ∆min = −1/8 and effective central charge ceff = 1.
(joint work with Paul A. Pierce)
