Painlevé equations and cellular automata

Summary

Abstract

We carry the discretisation of the Painlevé equations one step further by introducing systems that are cellular automata in the sense that the dependent variables take only integer values. This ultra-discretisation procedure is based on recent progress that led to a systematic way for the construction of ultra-discrete systems starting from known discrete forms. The ultra-discrete Painlevé equations have properties characteristic of the continuous and discrete Painlevé's, like coalescence cascades, particular solutions and auto-Backlund relations.

Introduction

Cellular automata are usually thought of as hyper-simplified evolution equations whereonly the fundamental features of the dynamics are retained. Painlevé equations are the simplest nontrivial integrable equations which encapsulate all the characteristics of integrable systems. This work will attempt to blend these two interesting ingredients and produce a new kind of systems: cellular automata that behave like Painlevé equations.

Why are cellular automata (CA) interesting? The key words are simplicity and richness.Since CA take values in a finite field (often {0,1}) or in ℤ, the numerical computations associated with their simulation are fast and reliable. Neither discretisatin nor truncation errors can perturb the accuracy of the result. On the other hand, studies of even the simplest CA have revealed a very rich dynamical behaviour. Even in the 1+1 dimensional case one finds behaviours ranging from integrability to chaos. While the use of CA in modelling generically chaotic equations has been extensive, the opposite is true as far as integrable equations are concerned. Relatively few studies exist to date on integrable cellular automata.