ULTRADISCRETE CONNECTION MATRICES OVER A TROPICAL SEMIRING

Chris Ormerod

Abstract

We consider linear problems associated with integrable ultra-discrete equations. In this paper, we study systems of linear difference equations over a tropical semiring. We prove that the fundamental solutions and monodromy matrix is well-defined for a class of such inear equations over the invertible max-plus algebra. We then proceed to extend the theory to define a funda- mental solution over the max-plus algebra. As an application we consider some special forms of the ultradiscrete Riccati equation and show it possesses a fundamental solution.

Keywords: Ultradiscrete, max-plus, monodromy, Integrable.

AMS Subject Classification: Primary 39A13; secondary 33E17, 37B15.