In 1997 Hirota introduced a discrete dynamical system dKdV that lies conceptually between the classical KdV equation (a PDE) and the box ball system (a cellular automaton). The most remarkable property of each of these systems is the existence of so-called soliton solutions. I will describe a new variant on dKdV and explain what its solitons look like. The system is built out of reduced word dynamics in the affine symmetric group, with the generators enriched by weights that transform according to the Lusztig relation (a,b,c) \mapsto (bc/(a+c), a+c, ab/(a+c)). This is joint work with P. Pylyavskyy. Speaker(s): Max Glick (U. Connecticut)