Abstracts

Darboux chain in quantum and classical mechanics

Alexander Veselov (Loughborough University)

I will show that a special reduction of period-one closure of the Darboux chain for the Schrödinger operators with matrix potentials coincides with the Brun-Bogoyavlenskij equations of motion of the rigid body about centre of mass in the Newtonian field with a quadratic potential.  The corresponding matrix Schrödinger operators are maximally finite-gap (in some precise sense) with the spectrum explicitly described. The general 2-by-2-matrix case, containing some exotic matrix versions of the harmonic oscillator, will be discussed in more detail. 

The talk is based on a recent joint work with V.E. Adler.

Universal Methods for Computing Spectra of Linear and Nonlinear Eigenvalue Problems

Catherine Drysdale (Lancaster University)

In this talk, I will discuss the algorithms and corresponding assumptions needed to compute spectra for linear and nonlinear problems. These algorithms ensure that a spectrum is calculated without spectral pollution or spectral invisibility for particular classes of operators. The mathematical machinery needed for nonlinear problems is a natural extension to what is needed for the linear eigenvalue problems. I will show the computed spectrum some key examples including the Imaginary Cubic Oscillator and the Klein Gordon equation.

C-integrable models in statistical mechanics

Francesco Giglio (University of Glasgow)

Abstract: Recent studies show that a class of models in equilibrium statistical mechanics and thermodynamics can be formulated in terms of c-integrable partial differential equations of hydrodynamic type for the associated order parameters. These are nonlinear equations that can be mapped to linear ones via nonlinear transformations, typically of Cole-Hopf type. Such models share common features across the finite-size regime, when a finite number of particles N is considered, the asymptotic regime of large (but finite) N, and the thermodynamic limit, where phase transitions manifest through the formation of viscous shocks.

The talk discusses this class of models, focusing on recent results (Phys. Rev. E 113, L042103 (2026)) concerning systems whose internal energy admits a virial-type expansion in terms of a single order parameter. We show that these models are exactly solvable to arbitrary expansion order: expectation values are determined by solutions of nonlinear c-integrable PDEs equipped with an initial datum, which can be computed explicitly from the partition function of non-interacting particles. In particular, we show that the free energy satisfies a nonlinear PDE involving Bell polynomials, and that the scaling behaviour near critical points is consistent with the Universality conjecture for viscous PDEs.

The formalism is applied to nuclear matter as a case study, where the nonlinear structure of the interaction and of the associated PDE leads to the emergence of multiple critical points. The virial coefficients can be tuned to reproduce both the hadron gas–quark-gluon and nuclear liquid–gas phase transitions at different energy scales, consistently with QCD predictions. Finally, we show how finite-N solutions provide a finite-size analogue of phase diagrams and introduce analytical tools to detect remnants of criticality when finite-size corrections are included.

The role of Toda integrals in the Fermi-Pasta-Ulam-Tsingou model

Helen Christodoulidi (University of Lincoln)

In recent years, there has been a surge in investigating the enigmatic persistence of non-equilibrium phenomena in many-body Hamiltonian systems. This talk aims to demonstrate that tools and methodologies derived from integrable systems theory can provide an ideal framework for studying Hamiltonian systems with many degrees of freedom, even at high energies or more generally under large perturbations, where stable integrable-like structures cease to exist. We will focus on the ergodic or non-ergodic properties of the Fermi-Pasta-Ulam-Tsingou (FPUT) model and its relevance to the integrable Toda lattice. In particular, we discuss the role of Toda integrals in identifying and measuring phase space diffusion in the FPUT model, as well as in estimating the equilibrium timescales. Finally, we will discuss the existence of a critical energy density, below which trajectories behave as in a Kolmogorov–Arnold–Moser (KAM) regime.

Macroscopic Integrability of the Exclusion Process

Kirone Mallik (IPhT CEA Saclay)

The exclusion process, considered as a paradigm to describe classical transport in non-equilibrium statistical physics,  is exactly solvable, microscopically.  Many analytical results have been derived by using `quantum integrability' techniques such as the Bethe Ansatz.  At the macroscopic scale, the behaviour of this system follows a variational principle, proposed by G. Jona-Lasinio and his collaborators,   known as the Macroscopic Fluctuation Theory (MFT). Optimal fluctuations far from equilibrium are determined by two coupled non-linear PDEs with mixed and non-local boundary conditions.

In this talk, we shall show that, for the exclusion process, the MFT system is classically integrable in the sense of  Liouville and can be analyzed with the help of the inverse scattering method.  By solving exactly the associated Riemann-Hilbert problem, we determine the large deviation function of the current and the optimal evolution that generates a required fluctuation, both at initial and final times.

Systems with Painlevé transcendents in the coefficients free of movable critical points

Marta Dell'Atti (University of Warsaw)

We construct a generalisation of what we call Bureau-Guillot systems: systems of first order equations with coefficient functions being Painlevé transcendents. The same Painlevé equation is related to the system and it appears as a regularising condition in the regularisation process. We extend the results of Bureau-Guillot considering polynomial systems free of movable critical points with degree larger than 2. These systems contain not only transcendents P-I and P-II in the coefficients, but also transcendents P-III, P-IV, P-V and P-VI (and/or their derivatives). We also present a simpler version of the change of variables to construct the analogues of the Bureau-Guillot systems in the "mixed" case: systems related to the equation P-J, with J=I, ..., VI, containing coefficient functions that are solutions to P-K with P-K ≠ P-J. 

The talk is based on the joint work with G. Filipuk [1,2]. 

[1] M. Dell'Atti, G. Filipuk: Quadratic Bureau-Guillot systems with the first and second Painlevé transcendents in the coefficients. Part I: geometric approach and birational equivalence, Results Math. (2026)

[2] M. Dell'Atti, G. Filipuk: Generalisation of Bureau-Guillot systems with Painlevé transcendents in the coefficients, Preprint (2026)

Hamiltonian Structure of Non-Homogeneous Systems: Geometry, Classifications, and Integrability

Pierandrea Vergallo (Università degli Studi della Basilicata)

This talk investigates the geometric and algebraic properties of non-homogeneous Hamilton- ian operators, defined as the sum of a first-order Dubrovin-Novikov operator and an ultralocal Poisson tensor. Starting from the key example of the Korteweg-de Vries equation, we explore the classification, compatibility, and application of these structures to quasilinear systems of PDEs which are non-homogeneous themselves.

We first present a complete classification of these operators, distinguishing between the non- degenerate case – characterized in Darboux coordinates by Lie algebras and quadratic Casimirs – and the degenerate case, providing a full description for operator with two and three components. These results are particularly relevant for inverted Hamiltonian systems where the leading term often loses rank.

The analysis then extends to the Hamiltonian formulation of quasilinear systems, where we identify the necessary geometric conditions using the theory of differential coverings. In this context, we present an extension of Tsarevs compatibility conditions to include operators with degenerate leading coefficients.

Finally, we discuss the integrability of these systems by establishing tensorial criteria for bi- Hamiltonianity and relating these operators to Nijenhuis geometry and Lie algebra frameworks.

This is based on [2, 3, 4] and is a joint work with Marta Dell’Atti and Alessandra Rizzo.

[1] Dubrovin B A, Novikov S P: On Poisson brackets of the hydrodynamic type. Akad. Nauk SSSR Dokl. 279:2 pp. 294–297 (1984).

[2] Dell’Atti M, Vergallo P: Classification of degenerate non-homogeneous Hamiltonian operators. J. Math. Phys. 64:3 (2022). 

DOI: https://doi.org/10.1063/5.0135134

[3] Vergallo P.: Non-homogeneous Hamiltonian structures for quasilinear systems. Boll. Unione Mat. Ital., pp. 114 (2024). 

DOI: https://doi.org/10.1007/s40574-023-00369-5

[4] Dell’Atti M, Rizzo A, Vergallo P: Geometric aspects of non-homogeneous 1+0 operators, arxiv: 2503.21917

On Fredholm Pfaffians and Riemann-Hilbert problems

Thomas Bothner (University of Bristol)

We will see how classes of Fredholm Pfaffians can be computed in terms of canonical, aux- iliary Riemann-Hilbert problems as soon as the main kernel in the Pfaffian is either of additive Hankel composition or of truncated Wiener-Hopf type. Akhiezer-Kac asymptotic results for the Fredholm Pfaffians will then be derived as natural consequences of the Riemann-Hilbert characterisation. 

Based on arXiv:2511.23362, joint work with Amari Jaconelli (Bristol).