Anton Dzhamay

On the Lagrangian Structure of the Discrete Isospectral and Isomonodromic Transformations

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, of two elementary factors. In this case, we establish, by explicitly writing down the Lagrangian, that the isomonodromic dynamic is Lagrangian. Next, we show how to make this Lagrangian time dependent to obtain the equations of the isomonodromic dynamic. In some special cases, such equations are known to reduce to the difference Painleve equations. We show how to obtain the difference Painleve V equation in that way, establishing that dPV can be written in the Lagrangian form.

Factorizations of rational matrix functions with application to discrete isomonodromic transformations and difference Painlevé equations

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 theory of discrete isomonodromic transformations and their relationship with difference Painleve equations. In particular, in these coordinates, basic isomonodromic transformations take the form of the discrete Euler–Lagrange equations. Secondly we show that dPV equations, previously obtained in this context by D Arinkin and A Borodin, can be understood as simple relationships between the residues of such matrices and their inverses.

Fiber-dependent deautonomization of integrable 2D mappings and discrete Painlevé equations

It is well known that two-dimensional mappings preserving a rational elliptic fibration, like the Quispel–Roberts–Thompson mappings, can be deautonomized to discrete Painleve 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 performing the deautonomization for a pair of an autonomous mapping and a fiber. Starting from a single autonomous mapping but varying the type of a chosen fiber, we obtain different types of discrete Painleve equations using this deautonomization procedure. We also introduce a technique for reconstructing a mapping from the knowledge of its induced action on the Picard group and some additional geometric data. This technique allows us to obtain factorized expressions of discrete Painleve equations, including the elliptic case. Further, by imposing certain restrictions on such non-autonomous mappings we obtain new and simple elliptic difference Painleve equations, including examples whose symmetry groups do not appear explicitly in Sakais classification.

COMBINATORICS OF MATRIX FACTORIZATIONS AND INTEGRABLE SYSTEMS

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 eigenvectors, edges are labeled by differences of locations of zeroes and poles of the determinant of our matrix function, and each face corresponds to a particular choice of a coordinate system on the space of such functions. Moreover, for each face this labeling encodes, in a neat and efficient way, a generating function for the expressions of the remaining four eigenvectors that label the opposing face of the cube in terms of the coordinates represented by the chosen face. The main motivation behind this work is that when our matrix is a Lax matrix of a discrete integrable system, such generating functions can be interpreted as Lagrangians of the system, and a choice of a particular face corresponds to a choice of the direction of the motion.