Discrete-time analysis for the integrable discrete Toda equations and the discrete Lotka-Volterra system

TL;DR

This paper derives general solutions and analyzes the long-term behavior of the discrete Toda equations and the discrete Lotka-Volterra system, revealing their asymptotic properties using novel analytical techniques.

Contribution

It provides the first simultaneous derivation of general solutions and asymptotic analysis for these important integrable discrete systems.

Findings

Explicit general solutions for both systems

Asymptotic behavior characterized for all initial conditions

Introduction of new techniques involving discrete-time variables and determinants

Abstract

The discrete autonomous/non-autonomous Toda equations and the discrete Lotka-Volterra system are important integrable discrete systems in fields such as mathematical physics, mathematical biology and statistical physics. They also have applications to numerical linear algebra. In this paper, we first simultaneously obtain their general solutions. Then, we show the asymptotic behavior of the solutions for any initial values as the discrete-time variables go to infinity. Our two main techniques for understanding the distinct integrable systems are to introduce two types of discrete-time variables and to examine properties of a restricted infinite sequence, its associated determinants and polynomials.