Solutions to the ultradiscrete Toda molecule equation expressed as minimum weight flows of planar graphs

TL;DR

This paper introduces a method using minimum weight flows on planar graphs to solve the ultradiscrete Toda molecule equation and related equations, highlighting a fundamental approach to their integrability.

Contribution

It presents a novel graph-based framework for solving ultradiscrete soliton equations, establishing a connection between minimum weight flows and integrability.

Findings

Function defined via minimum weight flow solves ultradiscrete Toda equations

Method applies to Bäcklund and two-dimensional Toda equations

Highlights fundamental approach to ultradiscrete soliton integrability

Abstract

We define a function by means of the minimum weight flow on a planar graph and prove that this function solves the ultradiscrete Toda molecule equation, its B\"acklund transformation and the two dimensional Toda molecule equation. The method we employ in the proof can be considered as fundamental to the integrability of ultradiscrete soliton equations.