Discrete solitons in infinite reduced words

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.