On the extension of the Painlevé property to difference equations

Abstract

It is well known that the integrability (solvability) of a differential equation is related to the singularity structure of its solutions in the complex domain - an observation that lies behind the Painlevé test. A number of ways of extending this philosophy to discrete equations are explored. First, following the classical work of Julia, Birkhoff and others, a natural interpretation of these equations in the complex domain as difference or delay equations is described and it is noted that arbitrary periodic functions play an analogous role for difference equations to that played by arbitrary constants in the solution of differential equations. These periodic functions can produce spurious branching in solutions and are factored out of the analysis which concentrates on branching from other sources. Second, examples and theorems from the theory of difference equations are presented which show that, modulo these periodic functions, solutions of a large class of difference equations are meromorphic, regardless of their integrability. It is argued that the integrability of many difference equations is related to the structure of their solutions at infinity in the complex plane and that Nevanlinna theory provides many of the concepts necessary to detect integrability in a large class of equations. A perturbative method is then constructed and used to develop series in z and the derivative of log(z), where z is the independent variable of the difference equation. This method provides an analogue of the series developed in the Painlevé test for differential equations. Finally, the implications of these observations are discussed for two tests which have been studied in the literature regarding the integrability of discrete equations.