NONLINEAR NONAUTONOMOUS DISCRETE DYNAMIC-SYSTEMS FROM A GENERAL DISCRETE ISOMONODROMY PROBLEM

Nonlinear nonautonomous discrete dynamical systems (DDS) whose continuum limits are the well-known Painleve equations, have recently arisen in models of quantum gravity. The Painleve equations are believed integrable because each is the isomonodromy condition for an associated linear differential equation. However, not every DDS with an integrable continuum limit is necessarily integrable. Which of the many discrete versions of the Painleve equations inherit their integrability is not known. How to derive all their integrable discrete versions is also not known. We provide a systematic method of attacking these questions by giving a general discrete isomonodromy problem. Discrete versions of the first and second Painleve equations are deduced from this general problem.