Discrete solitons in infinite reduced words

TL;DR

This paper models a discrete dynamical system using affine Weyl groups, interpreting it through wiring diagrams and $ au$-functions, and derives $N$-soliton solutions via reductions of the Hirota bilinear difference equation.

Contribution

It introduces a novel discrete system based on affine Weyl groups and provides explicit $N$-soliton solutions through a reduction of the Hirota equation.

Findings

System modeled using affine Weyl groups and wiring diagrams.

Realization of the system as a reduction of Hirota's bilinear difference equation.

Explicit construction of $N$-soliton solutions.

Abstract

We consider a discrete dynamical system where the roles of the states and the carrier are played by translations in an affine Weyl group of type $A$. The Coxeter generators are enriched by parameters, and the interactions with the carrier are realized using Lusztig's braid move $(a,b,c) \mapsto (bc/(a+c), a+c, ab/(a+c))$. We use wiring diagrams on a cylinder to interpret chamber variables as $\tau$-functions. This allows us to realize our systems as reductions of the Hirota bilinear difference equation and thus obtain $N$-soliton solutions.