Rational solutions to dPIV

Summary

Abstract

We study the rational solutions of the discrete version of Painleve's fourth equation (d-P_IV). The solutions are generated by applying Schlesinger transformations on the seed solutions −2z and −1/z. After studying the structure of these solutions we are able to write them in a determinantal form that includes an interesting parameter shift that vanishes in the continuous limit.

Introduction

One important question in the study of discrete versions of continuous differential equations concerns the existence of corresponding special solutions. For continuous Painlevé equations rational and special function solutions are known, and in many cases even a rigorous classification has been done. If one proposes a discrete version of a Painlevé equation it is not enough that in some continuous limit the continuous Painlevé equation is obtained, but in addition the proposed equation should share some further properties of the original equation. One of these properties should be the equivalent of the Painlevé property, called “singularity confinement”. This has already been used to propose discrete forms of the Painlevé equations. Other structures of the continuous Painlevé equations that have been shown to exist for the discrete ones include their relationships by coalescence limits and the existence of Hirota forms for these equations. What is still largely an open question is the fate of the special solutions (rational, algebraic, special function) known for the continuous case.