The general theory of linear difference equations over the invertible max-plus algebra

Nalini Joshi, Chris Ormerod

Abstract

We present the mathematical theory underlying systems of linear difference equations over the invertible max-plus algebra. The result provides an analogue of isomonodromy theory for ultradiscrete Painlevé equations, which are extended cellular automata, and provide evidence for their integrability. Our theory is analogous to that developed by Birkhoff and his school for q-difference linear equations but stands independently of the latter. As an example we derive linear problems in this algebra for ultradiscrete versions of the symmetric P_IV equation and show how it acts as the isomonodromic deformation of the linear system.

Keywords: Integrable, Cellular Automata, Tropical.

AMS Subject Classification: Primary 39A20,14H70,16Y60.