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

TL;DR

This paper investigates the complex dynamics of two identical in-phase bouncing droplets on a vibrating liquid surface, revealing diverse behaviors and bifurcations through theoretical and numerical analysis within a pilot-wave framework.

Contribution

It introduces a detailed theoretical and numerical study of two interacting bouncing droplets, highlighting new dynamical regimes and bifurcations in a pilot-wave system.

Findings

Droplets often form tightly bound pairs traveling together.

Bound pairs exhibit various trajectories including straight, oscillatory, and chaotic.

The system shows rich bifurcation behavior depending on parameters.

Abstract

A droplet bouncing on the surface of a vibrating liquid bath can move horizontally guided by the wave it produces on impacting the bath. The wave itself is modified by the environment, and thus the interactions of the moving droplet with the surroundings are mediated through the wave. This forms an example of a pilot-wave system. Taking the Oza Rosales Bush description for walking droplets as a theoretical pilot-wave model, we investigate the dynamics of two interacting identical, in-phase bouncing droplets theoretically and numerically. A remarkably rich range of behaviors is encountered as a function of the two system parameters, the ratio of inertia to drag, \k{appa}, and the ratio of wave forcing to drag, \b{eta}. The droplets typically travel together in a tightly bound pair, although they unbind when the wave forcing is large and inertia is small or inertia is moderately large and wave forcing is moderately small. Bound pairs can exhibit a range of trajectories depending on parameter values, including straight lines, sub-diffusive random walks, and closed loops. The droplets themselves may maintain their relative positions, oscillate towards and away from one another, or interchange positions regularly or chaotically as they travel. We explore these regimes and others and the bifurcations between them through analytic and numerical linear stability analyses and through fully nonlinear numerical simulation.