Anton Dzhamay

University of Northern Colorado | UNC · School of Ma

It is well-known that differential Painlevé equations can be written in a Hamiltonian form. However, a coordinate form of such representation is far from unique — there are many very different Hamiltonians that result in the same differential Painlevé equation. Recognizing a Painlevé equation, for example when it appears in some applied problem, is...

In this paper we consider two examples of certain recurrence relations, or nonlinear discrete dynamical systems, that appear in the theory of orthogonal polynomials, from the point of view of Sakai's geometric theory of Painlev\'e equations. On one hand, this gives us new examples of the appearance of discrete Painlev\'e equations in the theory of...

In this paper, we present a general scheme for how to relate differential equations for the recurrence coefficients of semiclassical orthogonal polynomials to the Painlevé equations using the geometric framework of the Okamoto Space of Initial Conditions. We demonstrate this procedure in two examples. For semiclassical Laguerre polynomials appearin...

It is well-known that differential Painlev\'e equations can be written in a Hamiltonian form. However, a coordinate form of such representation is far from unique -- there are many very different Hamiltonians that result in the same differential Painlev\'e equation. In this paper we describe a systematic procedure of finding changes of coordinates...

In this paper we present a general scheme for how to relate differential equations for the recurrence coefficients of semi-classical orthogonal polynomials to the Painlev\'e equations using the geometric framework of Okamoto's space of initial values. We demonstrate this procedure in two examples. For semi-classical Laguerre polynomials appearing i...

In this paper we present a connection between systems of differential equations for the recurrence coefficients of polynomials orthogonal with respect to the generalized Meixner and the deformed Laguerre weights. It is well-known that the recurrence coefficients of both generalized Meixner and deformed Laguerre orthogonal polynomials can be express...

Recurrence coefficients of semi-classical orthogonal polynomials are often related to the solutions of special nonlinear second-order differential equations known as the Painlevé equations. Each Painlevé equation can be written in a standard form as a non-autonomous Hamiltonian system, so it is natural to ask whether differential systems satisfied...

Over the last decade it has become clear that discrete Painlevé equations appear in a wide range of important mathematical and physical problems. Thus, the question of recognizing a given non-autonomous recurrence as a discrete Painlevé equation and determining its type according to Sakai's classification scheme, understanding whether it is equival...

In this paper we study a certain recurrence relation, that can be used to generate ladder operators for the Laguerre unitary ensemble, from the point of view of Sakai’s geometric theory of Painlevé equations. On one hand, this gives us one more detailed example of the appearance of discrete Painlevé equations in the theory of orthogonal polynomials...

In this paper we study a certain recurrence relation, that can be used to generate ladder operators for the Laguerre unitary ensemble, from the point of view of Sakai’s geometric theory of Painlev ́e equations. On one hand, this gives us one more detailed example of the appearance of discrete Painleve equations in thetheory of orthogonal polynomial...

In this paper we study a certain recurrence relation, that can be used to generate ladder operators for the Laguerre Unitary ensemble, from the point of view of Sakai's geometric theory of Painlev\'e equations. On one hand, this gives us one more detailed example of the appearance of discrete Painlev\'e equations in the theory of orthogonal polynom...

The goal of this paper is to investigate the missing part of the story about the relationship between the orthogonal polynomial ensembles and Painlevé equations. Namely, we consider the $q$-Racah polynomial ensemble and show that the one-interval gap probabilities in this case can be expressed through a solution of the discrete $q$-P$\left (E_7^{(1...

Over the last decade it has become clear that discrete Painlev\'e equations appear in a wide range of important mathematical and physical problems. Thus, the question of recognizing a given non-autonomous recurrence as a discrete Painlev\'e equation and determining its type according to Sakai's classification scheme, understanding whether it is equ...

The goal of this paper is to investigate the missing part of the story about the relationship between the orthogonal polynomial ensembles and Painlev\'e equations. Namely, we consider the case of $q$-Racah polynomial ensemble and show that the one-interval gap probabilities in this case can be expressed through a solution of the discrete $q$-P$\lef...

Although the theory of discrete Painlevé (dP) equations is rather young, more and more examples of such equations appear in interesting and important applications. Thus, it is essential to be able to recognize these equations, to be able to identify their type, and to see where they belong in the classification scheme. The definite classification s...

Although the theory of discrete Painlev\'e (dP) equations is rather young, more and more examples of such equations appear in interesting and important applications. Thus, it is essential to be able to recognize these equations, to be able to identify their type, and to see where they belong in the classification scheme. The definite classification...

It is well known that two-dimensional mappings preserving a rational elliptic fibration, like the Quispel-Roberts-Thompson mappings, can be deautonomized to discrete Painlev\'e equations. However, the dependence of this procedure on the choice of a particular elliptic fiber has not been sufficiently investigated. In this paper we establish a way of...

We present two examples of reductions from the evolution equations describing discrete Schlesinger transformations of Fuchsian systems to difference Painlev\'e equations: difference Painlev\'e equation d-$P\left({A}_{2}^{(1)*}\right)$ with the symmetry group ${E}^{(1)}_{6}$ and difference Painlev\'e equation d-$P\left({A}_{1}^{(1)*}\right)$ with th...

Schlesinger transformations are algebraic transformations of a Fuchsian system that preserve its monodromy representation and act on the characteristic indices of the system by integral shifts. One of the important reasons to study such transformations is the relationship between Schlesinger transformations and discrete Painlev\'e equations; this i...

We study relations between the eigenvectors of rational matrix functions on the Riemann sphere. Our main result is that for a subclass of functions that are products of two elementary blocks it is possible to represent these relations in a combinatorial-geometric way using a diagram of a cube. In this representation, vertices of the cube represent...

We study factorizations of rational matrix functions with simple poles on the Riemann sphere. For the quadratic case (two poles) we show, using multiplicative representations of such matrix functions, that a good coordinate system on this space is given by a mix of residue eigenvectors of the matrix and its inverse. Our approach is motivated by the...

We study the Lagrangian properties of the discrete isospectral and isomonodromic dynamical systems. We generalize the Moser–Veselov approach to integrability of discrete isospectral systems via the refactorization of matrix polynomials to matrix rational functions with a simple divisor, and consider in detail the case of two poles or, equivalently,...

In this paper we present a construction of a new class of explicit solutions to the WDVV (or associativity) equations. Our construction is based on a relationship between the WDVV equations and Whitham (or modulation) equations. Whitham equations appear in the perturbation theory of exact algebro-geometric solutions of soliton equations and are def...

Non-classical infinitesimal weak symmetries of PDE introduced by Olver and Rosenau (1987) are analysed. In the case of PDE in two independent variables, it is demonstrated that obtaining such symmetries is equivalent to obtaining the two-dimensional modules of non-classical partial symmetries. The Boussinesq and the nonlinear heat equations are tre...