Bilinear Discrete Painleve-II and its Particular Solutions

Abstract

By analogy to the continuous Painlevé II equation, we present particular solutions of the discrete Painlevé II (d-P$\rm_{II}$) equation. These solutions are of rational and special function (Airy) type. Our analysis is based on the bilinear formalism that allows us to obtain the $τ$ function for d-P$\rm_{II}$. Two different forms of bilinear d-P$\rm_{II}$ are obtained and we show that they can be related by a simple gauge transformation.