An Ultradiscrete Matrix Version of the Fourth Painlevé Equation

Abstract

This paper is concerned with the matrix generalization of ultradiscrete systems. Specifically, we establish a matrix generalization of the ultradiscrete fourth Painlevé equation (ud-P IV ). Well-defined multicomponent systems that permit ultradiscretization are obtained using an approach that relies on a group defined by constraints imposed by the requirement of a consistent evolution of the systems. The ultradiscrete limit of these systems yields coupled multicomponent ultradiscrete systems that generalize ud-P IV . The dynamics, irreducibility, and integrability of the matrix-valued ultradiscrete systems are studied.