Discrete Schlesinger Equations and Difference Painlevé Equations.
Anton Dzhmay
School of Mathematical Sciences, University of Northern Colorado
Date and time:
Tuesday, March 4, 2014 - 4:30pm
Location:
ECOT 226
Abstract:
The theory of Schlesinger equations describing isomonodromic dynamic on the space of matrix coefficients of a Fuchsian system w.r.t. continuous deformations is well-know. In this talk we consider a discrete version of this theory. Discrete analogues of Schlesinger deformations are Schlesinger transformations that shift the eigenvalues of the coefficient matrices by integers. By discrete Schlesinger equations we mean the evolution equations on the matrix coefficients describing such transformations. We derive these equations, show how they can be split into the evolution equations on the space of eigenvectors of the coefficient matrices, and explain how to write the latter equations in the discrete Hamiltonian form. We also consider some reductions of those equations to the difference Painlevé equations, again in complete parallel to the differential case.
This is a joint work with T.Takenawa (Tokyo Institute of Marine Science and Technology) and H. Sakai (the University of Tokyo)