Abstracts

Towards a 2-dimensional generalised hydrodynamics

Thibault Bonnemain (King's College London)

Generalised hydrodynamics is a recent and powerful framework to study many-body integrable systems, quantum or classical, out of equilibrium. It has been applied to several models, from delta Bose gas to the XXZ spin chain, the KdV soliton gas and many more... yet it has only been applied to (1+1)D systems. This theory is in particular predicated on the fact that the scattering of the quasi-particles of the microscopic model is 2-bodyfactorisable (or at least that 3 or higher body interactions can be neglected), which is true for integrable systems and generally a reasonable assumption in (1+1)D. Using simple geometric arguments I will try to propose an extension of GHD to a 2D toy model featuring line solitons, inspired by the phenomenology of the KP equation and on the recently established connection between GHD and soliton gas theory.

This presentation is based on preliminary work with Benjamin Doyon.

Entanglement Content of Localised Excitations: Symmetry Resolution

Olalla Castro Alvaredo (City University London)

In this talk I will introduce some basic ideas about entanglement measures in many-body quantum systems and I will present one of the leading approaches to computing such measures in 1D quantum field theory. This approach is based on relating entanglement measures to correlations functions of a special class of fields called branch point twist fields. Once this connection has been made, the problem of computing entanglement measures is reduced to computing correlation functions, which is generally technically difficult. I will explain how these correlation functions become especially simple for certain types of excited states of quantum field theory and how this simplicity allows us to compute many different measures very explicitly, including a measure that has attracted a lot of interest recently: the symmetry resolved entanglement entropy.

My talk contains input from work I have done with many people over the years including Pasquale Calabrese, Luca Capizzi, John L. Cardy, Cecilia De Fazio, Benjamin Doyon, David X. Horváth, Michele Mazzoni, Lucía Santamaría-Sanz and István Szécsény.

Lagrangian multiforms: a variational criterion for integrability

Vincent Caudrelier (University of Leeds)

I will present the relatively recent notion of Lagrangian multiforms whose aim is to capture integrability in a purely variational fashion. Lagrangian multiforms are ubiquitous in classical integrable models: they can be defined and used in continuous or discrete systems, finite or infinite dimensional (field theories). So far, the Hamiltonian formalism has been the overwhelming winner to define integrability, rooted in the Liouville-Arnold theorem. I will show how Lagrangian multiforms offer variational counterparts to the established Hamiltonian criteria for integrability. For instance, the so-called "closure relation" on the multiform corresponds to Poisson commuting Hamiltonians and can be linked to the classical Yang-Baxter equation. This continues the long interplay between Hamiltonian and Lagrangian formalisms, within the realm of integrability. I will present various examples of construction of Lagrangian multiforms for field theories (AKNS hierarchy) and finite dimensional systems (Toda hierarchy) which exploit some of the standard machinery such as coadjoint orbits and classical r-matrix. I will conclude with some comments on one important motivation of this programme: construct a path integral alternative to the well-established, Hamiltonian based, Quantum Inverse Scattering Method framework.

Classification of degenerate non-homogeneous Hamiltonian operators

Marta Dell'Atti (University of Portsmouth)

We investigate non-homogeneous Hamiltonian operators of order 1+0, composed of a first order Dubrovin-Novikov operator and an ultralocal one. The study of such operators turns out to be relevant for the inverted system of equations associated with a class of Hamiltonian scalar equations, where the operator of order 1 is often degenerate. We derive a complete classification of the 1+0 operators with degenerate leading coefficient in systems with two and three components in 1+1 dimensions.

Quasilinear systems of Jordan block type

Jenya Ferapontov (Loughborough University)

Systems of Jordan block type have appeared recently in applications as reductions of multidimensional quasilinear PDEs and as reductions of the kinetic equation for soliton gas. The following topics will be discussed:

• Systems of Jordan block type and modified KP hierarchy;

• Hydrodynamic reductions of the kinetic equation for soliton gas;

• Hamiltonian aspects of systems of Jordan block type.

The talk is based on the following publications:
Lingling Xue, E.V. Ferapontov, Quasilinear systems of Jordan block type and the mKP hierarchy,  J. Phys. A: Math. Theor. 53 (2020) 205202 (14pp); https://doi.org/10.1088/1751-8121/ab859a; arXiv:2001.03601.
E.V. Ferapontov, M.V. Pavlov, Kinetic equation for soliton gas: integrable reductions,  J. Nonlinear Sci. 32 (2022), no. 2, Paper No. 26, 22 pp; https://doi.org/10.1007/s00332-022-09782-0;
P. Vergallo, E.V. Ferapontov, Hamiltonian systems of Jordan block type; arXiv:2212.01413.

Semiclassical orthogonal polynomials and the first discrete Painlevé hierarchy

Ana Loureiro (University of Kent)

The recurrence coefficients in the three-term recurrence relation of a polynomial sequence orthogonal with respect to a quartic, a sextic or higher order Freud weights are a solution of a forth order discrete equation which is a member of the first discrete Painlevé hierarchy. They also satisfy a coupled system of second-order, nonlinear differential equations. Such orthogonality weights also arise in the context of Hermitian matrix models and random symmetric matrix ensembles. In this talk I will report on the behaviour of such special function solutions and explain how the study may inform on the study of recurrence coefficients associated with higher order Freud weights. The emphasis will be on their asymptotic periodic properties.  

Collaborators: Peter Clarkson (University of Kent) and Kerstin Jordaan (University of South Africa)

Random matrices, orthogonal polynomials and dispersive hydrodynamic equations

Antonio Moro (Northumbria University)

We briefly review some key aspects of the well established mutual connections between the theory of random matrix ensembles, orthogonal polynomials and integrable systems. On this basis, we discuss how elements of emergent complexity in random matrix theory and orthogonal polynomials can be understood in terms of multiscale asymptotics and normal forms of integrable nonlinear PDEs.

Dispersive quantisation, linear and nonlinear

Beatrice Pelloni (Heriot-Watt University)

I will discuss a surprising phenomenon, first observed experimentally in linear optics and quantum wave transmission, and known variously as the Talbot effect, fractalisation, (quantum) revivals, or dispersive quantisation. This phenomenon is manifested in the solution of periodic dispersive equation, starting for a discontinuous initial condition. Then, at times that are rational multiple of the spatial period (“rational” times), the solution is discontinuous, indeed it is built from translated copies of the initial condition, while at all other times, the solution is wildly oscillatory, with positive fractal dimension, but it is continuous.

I will discuss the mathematical description of this phenomenon, and the effect on it of nonlinearity, integrability and non-periodic boundary conditions.