Advances in Difference Equations
Volume 2007 (2007), Article ID 96752, 15 pages
Research Article

An Ultradiscrete Matrix Version of the Fourth Painlevé Equation

Chris M. Field and Chris M. Ormerod

School of Mathematics and Statistics F07, The University of Sydney, Sydney NSW 2006, Australia

Received 27 February 2007; Accepted 1 May 2007

Academic Editor: Kilkothur Munirathinam Tamizhmani

Copyright © 2007 Chris M. Field and Chris M. Ormerod. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper is concerned with the matrix generalization of ultradiscrete systems. Specifically, we establish a matrix generalization of the ultradiscrete fourth Painlevé equation (ud-PIV). 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-PIV. The dynamics, irreducibility, and integrability of the matrix-valued ultradiscrete systems are studied.