Ultradiscrete Lotka-Volterra system computes tropical eigenvalue of symmetric tridiagonal matrices

Abstract

Abstract Some of authors’ recent study shows that the time evolution of the integrable ultradiscrete Toda equation computes eigenvalue of tridiagonal matrices over min-plus algebra, where min-plus algebra is a semiring with two binary operations: ⊕ : = min and ⊗ : = +. In this paper, we rst present a Backlund transformation between the ultradiscrete Toda equation and the ultradiscrete Lotka-Volterra system. Using the Backlund transformation, we show that the ultradiscrete Lotka-Volterra system can also compute eigenvalue of symmetric tridiagonal matrices over min-plus algebra.