# Nonautonomous ultradiscrete hungry Toda lattice and a generalized box-ball system

## TL;DR

This paper introduces a nonautonomous ultradiscrete hungry Toda lattice linked to a generalized box-ball system, providing new insights into integrable systems and their discrete solutions.

## Contribution

It derives a nonautonomous ultradiscrete Toda lattice with boundary conditions and connects it to a generalized box-ball system using discrete biorthogonal polynomials.

## Key findings

- Established a direct connection between ultradiscrete Toda lattice and box-ball system.
- Constructed particular solutions using discrete biorthogonal polynomials.
- Extended the understanding of nonautonomous integrable systems in discrete settings.

## Abstract

A nonautonomous version of the ultradiscrete hungry Toda lattice with a finite lattice boundary condition is derived by applying reduction and ultradiscretization to a nonautonomous two-dimensional discrete Toda lattice. It is shown that the derived ultradiscrete system has a direct connection to the box-ball system with many kinds of balls and finite carrier capacity. Particular solutions to the ultradiscrete system are constructed by using the theory of some sort of discrete biorthogonal polynomials.