  We present results of numerical simulations of the kinetics of
exciton-exciton annihilation of weakly localized one-dimensional Frenkel
excitons at low temperatures. We find that the kinetics is represented by two
well-distinguished components: a fast short-time decay and a very slow
long-time tail. The former arises from excitons that initially reside in states
belonging to the same localization segment of the chain, while the slow
component is caused by excitons created on different localization segments. We
show that the usual bi-molecular theory fails in the description of the
behavior found. We also present a qualitative analytical explanation of the
non-exponential behavior observed in both the short- and the long-time decay
components.
