How can a single structure have simultaneously the properties of a localized particle and those of a wave? This is the central question at the scale of elementary particles. Perfectly characterized in the formalism of quantum mechanics, this duality is usually thought to have no possible equivalent in classical physics. We were driven into revisiting this question by the following experiment. On the surface of a vertically vibrated liquid, a droplet can be kept bouncing for an indefinite time. This bouncing generates waves and the droplet can becomes self-propelled by its interaction with those waves. We thus obtain a single structure, a "walker", in which the drop generates the wave and the wave drives the drop's motion. Various experiments were undertaken to characterize their dynamical behaviors. Surprisingly quantum-like behaviours were observed to show-up. In this talk the motions of a walker submitted to either a transverse or a central force will be described and compared to the genuine quantum cases.

The wave field structure will be discussed. Close to the Faraday instability threshold the waves generated at each bounce are almost sustained. Therefore the global wave field results from the superposition of wave packets generated in the past. It thus contains both a memory of the drop trajectory as well as information on the medium boundaries. In our experiment this particular form of non-locality is responsible for the observed quantum-like behaviors.