Integrable Systems Related to Matrix LR Transformations

Abstract

The discrete Toda (dToda) equation, which is a representative integrable system, is the recursion formula of the well-known quotient-difference algorithm for computing the eigenvalues of tridiagonal matrices. In other words, the dToda equation is related to the LR transformations of tridiagonal matrices. In this chapter, by extending the application of LR transformations from tridiagonal to Hessenberg matrices, we capture the discrete hungry Toda (dhToda) and discrete relativistic Toda (drToda) equations, which are extensions of the dToda equation from the perspective of LR transformations. From the LR perspective, we identify further extensions of the dhToda equations, and clarify the relationship between the drToda equation and the discrete hungry Lotka–Volterra system. We also demonstrate that ultradiscrete versions of discrete integrable systems related to the LR transformations can be used to compute the eigenvalues of matrices over min-plus algebra, as can discrete integrable systems over linear algebra. These ultradiscrete integrable systems are expected to be equations of motion for box-and-ball systems (BBSs), which are cellular automata that describe mobility phenomena. Thus, we present an example of utilizing the LR perspective to design a new BBS.