Programme: Analytic and Geometric Methods in Integrable Systems

Below is the programme of the workshop. The recorded talks are available here.

FRIDAY, MAY 14TH

2.30pm (UK time): Tamara Grava (Bristol and SISSA)

Title:  Gibbs ensemble for the Ablowitz-Ladik lattice,  circular beta-ensemble and double confluent Heun equation

Abstract: We consider the discrete nonlinear Schrödinger equation in its integrable version, that is called Ablowitz Ladik lattice. We consider the Generalized Gibbs ensemble for the Ablowitz Ladik lattice with periodic boundary conditions with period $N$. In this setting the Lax matrix of the Ablowitz Ladik is a random CMV-periodic matrix  that is related to the Killip-Nenciu circulant $\beta$-ensemble at high temperature. We obtain the generalized free energy of the Ablowitz-Ladik lattice and the density of states of the random Lax matrix by establishing a mapping to the one-dimensional log-gas with an interaction strength of order $1/N$. For the Gibbs measure related to the Hamiltonian of the Ablowitz-Ladik flow, we obtain the density of states via a particular solution of the double-confluent Heun equation.

3.45pm (UK time): Samuel Fromm (KTH, Stockholm)

Title: The defocusing nonlinear Schrödinger equation with step-like oscillatory initial data

Abstract: In this talk, we study the Cauchy problem for the defocusing nonlinear Schrödinger equation
under the assumption that the solution vanishes as $x \to \infty$ and approaches an oscillatory plane
wave as $x \to -\infty$. We show that this problem has a global solution and that this solution can be represented in terms of the solution of a Riemann–Hilbert problem. By performing a steepest descent analysis of this Riemann–Hilbert problem, we also derive asymptotic formulas for the solution. This is joint work with Jonatan Lenells and Ronald Quirchmayr. Paper reference: arXiv:2104.03714.

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SATURDAY, MAY 15TH

10.30am (UK time): Pavlos Kassotakis (Cyprus)

Title: Discrete Lax pairs and hierarchies of integrable difference systems

Abstract: We introduce a family of order $N\in \mathbb{N}$ Lax matrices that is indexed by the natural number $k\in \{1,\ldots,N-1\}.$ For each value of $k$, they serve as strong Lax matrices of a hierarchy of integrable difference systems in edge variables that in turn lead to hierarchies of integrable difference systems in vertex variables or in a combination of edge and vertex variables. Furthermore, the entries of the Lax matrices are considered as elements of a division ring, so we obtain  hierarchies of
discrete integrable systems extended in the non-commutative  domain.

11.45am (UK time): Panagiota-Maria Adamopoulou (Heriot-Watt)

Title: Integrable Yang-Baxter maps via Grassmann algebras

Abstract: I will discuss some recent work on integrable extensions via Grassmann algebras of known Yang-Baxter maps. First, I will describe how refactorisation problems associated to Darboux transformations of certain integrable PDEs lead to solutions of the set-theoretic Yang-Baxter equation (Yang-Baxter maps). Then, I will introduce certain extensions via Grassmann algebras of known maps of Yang-Baxter type. I will discuss some integrability properties (such as invariants, Lax representation) of these newly obtained maps and some open problems in relation to this. Joint work with S. Konstantinou-Rizos and G. Papamikos.