  We study chemically driven running droplets on a partially wetting solid
substrate by means of coupled evolution equations for the thickness profile of
the droplets and the density profile of an adsorbate layer. Two models are
introduced corresponding to two qualitatively different types of experiments
described in the literature. In both cases an adsorption or desorption reaction
underneath the droplets induces a wettability gradient on the substrate and
provides the driving force for droplet motion. The difference lies in the
behavior of the substrate behind the droplet. In case I the substrate is
irreversibly changed whereas in case II it recovers allowing for a periodic
droplet movement (as long as the overall system stays far away from
equilibrium). Both models allow for a non-saturated and a saturated regime of
droplet movement depending on the ratio of the viscous and reactive time
scales. In contrast to model I, model II allows for sitting drops at high
reaction rate and zero diffusion along the substrate. The transition from
running to sitting drops in model II occurs via a super- or subcritical
drift-pitchfork bifurcation and may be strongly hysteretic implying a
coexistence region of running and sitting drops.
