The stability of bound states in pilot-wave hydrodynamics

Abstract

Millimetric droplets bouncing on the surface of a vertically vibrating fluid bath may self-propel through a resonant interaction with their own wavefield, displaying behaviors previously thought to be exclusive to the microscopic quantum realm. We investigate the stability of quantized bound states comprised of multiple droplets interacting through their shared wavefield, using an integrated experimental and theoretical approach. We consider the behavior of droplet pairs, rings, and chains as the bath's vibrational acceleration is increased progressively, and uncover a rich variety of dynamical states including periodic oscillations and traveling waves. The instability observed is dependent on the droplet number and size, and whether the drops are bouncing in- or out-of-phase relative to their neighbors. We develop a new theoretical model that accounts for the coupling between a drop's horizontal and vertical motion, enabling us to rationalize the majority of our experimental findings. We thus demonstrate that variations in a drop's impact phase with the bath have a critical influence on the stability of bouncing-droplet bound states. Our work provides insight into the complex interactions and collective motions that arise in bouncing-droplet aggregates, and forges new mathematical links with extant models of microscopic physics.