On the monodromy of the deformed cubic oscillator

We study a second-order linear differential equation known as the deformed cubic oscillator, whose isomonodromic deformations are controlled by the first Painlevé equation. We use the generalised monodromy map for this equation to give solutions to the Riemann-Hilbert problems of (Bridgeland in Invent Math 216(1):69–124, 2019) arising from the Donaldson-Thomas theory of the A$$_2$$ 2 quiver. These are the first known solutions to such problems beyond the uncoupled case. The appendix by Davide Masoero contains a WKB analysis of the asymptotics of the monodromy map.

On the Question of the Bäcklund Transformations and Jordan Generalizations of the Second Painlevé Equation

We demonstrate the way to derive the second Painlevé equation P2 and its Bäcklund transformations from the deformations of the Nonlinear Schrödinger equation (NLS), all the while preserving the strict invariance with respect to the Schlesinger transformations. The proposed algorithm allows for a construction of Jordan algebra-based completely integrable multiple-field generalizations of P2 while also producing the corresponding Bäcklund transformations. We suggest calling such models the JP-systems. For example, a Jordan algebra JMat(N,N) with the Jordan product in the form of a semi-anticommutator is shown to generate an integrable matrix generalization of P2, whereas the VN algebra produces a different JP-system that serves as a generalization of the Sokolov’s form of a vectorial NLS.

Differential and difference equations for recurrence coefficients of orthogonal polynomials with hypergeometric weights and Bäcklund transformations of the sixth Painlevé equation

It is known from [G. Filipuk and W. Van Assche, Discrete orthogonal polynomials with hypergeometric weights and Painlevé VI, Symmetry Integr. Geom. Methods Appl. 14 (2018), Article ID: 088, 19 pp.] that the recurrence coefficients of discrete orthogonal polynomials on the nonnegative integers with hypergeometric weights satisfy a system of nonlinear difference equations. There is also a connection to the solutions of the [Formula: see text]-form of the sixth Painlevé equation (one of the parameters of the weights being the independent variable in the differential equation) [G. Filipuk and W. Van Assche, Discrete orthogonal polynomials with hypergeometric weights and Painlevé VI, Symmetry Integr. Geom. Methods Appl. 14 (2018), Article ID: 088, 19 pp.]. In this paper, we derive a second-order nonlinear difference equation from the system and present explicit formulas showing how this difference equation arises from the Bäcklund transformations of the sixth Painlevé equation. We also present an alternative way to derive the connection between the recurrence coefficients and the solutions of the sixth Painlevé equation.