The Kostant-Toda equation and the hungry integrable systems

Abstract

The Toda equation, which has applications in many fields, is one of the most important integrable systems. A generalization of the Toda equation is the Kostant-Toda equation, whose Lax representation is expressed using bidiagonal and lower Hessenberg matrices. The discrete Toda equation, which is a time-discretization of the Toda equation, is extended to the so-called discrete hungry Toda (dhToda) equation. The Lax representation for the dhToda equation generates similarity transformations of lower Hessenberg matrices. Although both have been considered, a discrete analogue of the Kostant-Toda equation or a continuous analogue of the dhToda equation has yet to be presented in the literature. In this paper, we present a continuous analogue of the dhToda equation, and then relate this to the Kostant-Toda equation. By using a continuous analogue of the Bäcklund transformation between the dhToda equation and the discrete hungry Lotka-Volterra system, we relate the resulting continuous analogue to the continuous hungry Lotka-Volterra system.