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A droplet bouncing on the surface of a vibrating liquid bath can movehorizontally guided by the wave it produces on impacting the bath. The waveitself is modified by the environment, and thus the interactions of the movingdroplet with the surroundings are mediated through the wave. This forms anexample of a pilot-wave system. Taking the Oza Rosales Bush description forwalking droplets as a theoretical pilot-wave model, we investigate the dynamicsof two interacting identical, in-phase bouncing droplets theoretically andnumerically. A remarkably rich range of behaviors is encountered as a functionof the two system parameters, the ratio of inertia to drag, \k{appa}, and theratio of wave forcing to drag, \b{eta}. The droplets typically travel togetherin a tightly bound pair, although they unbind when the wave forcing is largeand inertia is small or inertia is moderately large and wave forcing ismoderately small. Bound pairs can exhibit a range of trajectories depending onparameter values, including straight lines, sub-diffusive random walks, andclosed loops. The droplets themselves may maintain their relative positions,oscillate towards and away from one another, or interchange positions regularlyor chaotically as they travel. We explore these regimes and others and thebifurcations between them through analytic and numerical linear stabilityanalyses and through fully nonlinear numerical simulation.

Pilot-wave dynamics of two identical, in-phase bouncing droplet