Solitons in the one-dimensional forest fire model

The forest fire model is a simplified model of turbulent phenomena. Propagating fires dissipate or burn trees at an average rate determined by the tree growth probability, $p$, representing power fed into the system. The fires are self-sustaining, with no spontaneous ignition. In one-dimension fires propagate as solitons, resembling shocks in Burgers turbulence. The branching of solitons, creating new fires, is balanced by the pair-wise annihilation of oppositely moving solitons. Two diverging length scales appear in the slow driving limit, $p \to 0$. The width of the solitons, $w$, diverges as a power law, $1/p$, while the average distance between solitons exhibits an essential singularity, growing as $ \exp({\pi}^2/12p)$.