A sine-Gordon cellular automaton and its exotic solitons

We study the solitons of the ultradiscrete potential sine-Gordon equation and their interaction. We show that the collision of these solitons can be either elastic or inelastic. In the latter case, a soliton can emerge from the interaction either longer (dressed) or shorter (undressed), but there are upper and lower limits on the possible lengths of each soliton. Starting from the relation of the discrete sine-Gordon to the discrete KdV, we derive its ultradiscrete analogue and use it to transpose the study of the interaction of the sine-Gordon solitons to the dynamics of the solutions of KdV. The latter is represented by symbolic-like dynamics of the background term present in the ultradiscrete KdV equation. We introduce a new elementary entity, dubbed 'oiston', and describe these dynamics as the interaction of the latter. This allows us to explain the dressing and undressing of the sine-Gordon solitons as well as the fact that this process has a fixed capacity for every soliton. (paper)