Exactly Solvable and Integrable Systems Papers of the Week

At conjugate fugacities outside the unit circle, nearly square droplets produce coefficient growth incompatible with an algebraic differential equation. We prove that cancellations cannot remove this growth in the full pressure series.

We present a systematic study of the mathematical theory of the generalized Brézin-Gross-Witten (BGW) theory using methods from statistical physics.

We study the root-of-unity limit of the multi-component spin solution of the star-star relation whose Boltzmann weights are products of lens elliptic gamma functions, and whose spins take values in $\mathbb{R}^{n-1}\times(\mathbb{Z}_r)^{n-1}$.

Here, we present an elementary solution of the quantum Gérard-Grellier Hamiltonian.

We study rogue wave formation in the dam-break problem for the Kundu--Eckhaus equation with deterministic and randomly perturbed box initial data.

In this paper, we present a full Darboux transformation for the Lax triplet of the lpKdV equation, and eigenfunctions are formulated in terms of Casoratians.

We consider nearest-neighbour interactions on spin-1/2 chains. We find the full set of integrable models of this form and apply a numerical algorithm to determine the "closest" integrable model for a given Hamiltonian.

However, this has never been done for classical field theories: no classical derivation has consistently accounted for solitons, breathers, radiation (dispersive modes), and their mutual interactions. We here derive the TBA equations for the Toda chain directly from its inverse scattering transform.

With non-diagonal boundaries, determining these configurations remains challenging even when an exact spectral solution is available, hindering the direct extraction of thermodynamic properties. We address this problem for the rational $C_2^{(1)}$ chain by combining DMRG in the tensor-network framework with the quantum inverse scattering method.

We construct a discrete four-component massive Thirring model (MTM) within the Cauchy matrix framework.

Recently, a new understanding of topological recursion has emerged from $x$-$y$ duality, which relates two such families through explicit formulae. We study $x$-$y$ duality, its parametrised version, and symplectic duality in the setting of equivariant topological recursion.

The hard-hexagon model was solved in this way, while the neighbouring hard-square model has no known closed form.

We consider a system of two bilinear discrete equations with four parameters.

We introduce the notion of an oriented $\vee$-system which gives a class of solutions of the oriented associativity equations generalising Veselov's $\vee$-system solutions of WDVV equations.

We describe a symplectic reduction of the Zhukovskiy integrable case of motion of a rigid body about a fixed point with a gyroscope in $\mathbb R^n$ to an integrable magnetic flow on a sphere $S^{n-1}$.

Here, we prove a formula for the dimensional counting of the Poisson cohomology for arbitrary $D$.

As consequences, we describe the zero-mean ensemble, and we recover, term by term in the genus expansion, the polygon-gluing formula for the Gaussian Unitary Ensemble in the limit $c_A=c_B\to\infty$.

We study the two fourth-order quantum superintegrable systems in the Euclidean plane that separate in polar coordinates and whose angular potentials are determined by the general four-parameter sixth Painlevé transcendent.