Constructing solutions to the ultradiscrete Painlevé equations

We investigate the nature of particular solutions to the ultradiscrete Painlevé equations. We start by analysing the autonomous limit and show that the equations possess an explicit invariant which leads naturally to the ultradiscrete analogue of elliptic functions. For the ultradiscrete Painlevé equations II and III we present special solutions reminiscent of the Casorati determinant ones which exist in the continuous and discrete cases. Finally we analyse the discrete Painlevé equation I and show how it contains both the continuous and the ultradiscrete ones as particular limits.