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1. Bouncing phase variations in pilot-wave hydrodynamics and the stability of droplet pairs

   https://doi.org/10.1017/jfm.2019.293  

   Couchman, Miles M.; Turton, Sam E.; Bush, John W. (July 2019, Journal of Fluid Mechanics)

   We present the results of an integrated experimental and theoretical investigation of the vertical motion of millimetric droplets bouncing on a vibrating fluid bath. We characterize experimentally the dependence of the phase of impact and contact force between a drop and the bath on the drop’s size and the bath’s vibrational acceleration. This characterization guides the development of a new theoretical model for the coupling between a drop’s vertical and horizontal motion. Our model allows us to relax the assumption of constant impact phase made in models based on the time-averaged trajectory equation of Moláček and Bush ( J. Fluid Mech. , vol. 727, 2013b, pp. 612–647) and obtain a robust horizontal trajectory equation for a bouncing drop that accounts for modulations in the drop’s vertical dynamics as may arise when it interacts with boundaries or other drops. We demonstrate that such modulations have a critical influence on the stability and dynamics of interacting droplet pairs. As the bath’s vibrational acceleration is increased progressively, initially stationary pairs destabilize into a variety of dynamical states including rectilinear oscillations, circular orbits and side-by-side promenading motion. The theoretical predictions of our variable-impact-phase model rationalize our observations and underscore the critical importance of accounting for variability in the vertical motion when modelling droplet–droplet interactions. 

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2. Pilot-wave dynamics: Using dynamic mode decomposition to characterize bifurcations, routes to chaos, and emergent statistics

   https://doi.org/10.1103/PhysRevE.108.034213  

   Kutz, J Nathan; Nachbin, André; Baddoo, Peter J; Bush, John_W M (September 2023, Physical Review E)

   We develop a data-driven characterization of the pilot-wave hydrodynamic system in which a bouncing droplet self-propels along the surface of a vibrating bath. We consider drop motion in a confined one-dimensional geometry and apply the dynamic mode decomposition (DMD) in order to characterize the evolution of the wave field as the bath’s vibrational acceleration is increased progressively. Dynamic mode decomposition provides a regression framework for adaptively learning a best-fit linear dynamics model over snapshots of spatiotemporal data. Thus, DMD reduces the complex nonlinear interactions between pilot waves and droplet to a low-dimensional linear superposition of DMD modes characterizing the wave field. In particular, it provides a low-dimensional characterization of the bifurcation structure of the pilot-wave physics, wherein the excitation and recruitment of additional modes in the linear superposition models the bifurcation sequence. This DMD characterization yields a fresh perspective on the bouncing-droplet problem that forges valuable new links with the mathematical machinery of quantum mechanics. Specifically, the analysis shows that as the vibrational acceleration is increased, the pilot-wave field undergoes a series of Hopf bifurcations that ultimately lead to a chaotic wave field. The established relation between the mean pilot-wave field and the droplet statistics allows us to characterize the evolution of the emergent statistics with increased vibrational forcing from the evolution of the pilot-wave field. We thus develop a numerical framework with the same basic structure as quantum mechanics, specifically a wave theory that predicts particle statistics. 

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3. Dynamics, emergent statistics, and the mean-pilot-wave potential of walking droplets

   https://doi.org/10.1063/1.5030639  

   Durey, Matthew; Milewski, Paul_A; Bush, John_W_M (September 2018, Chaos: An Interdisciplinary Journal of Nonlinear Science)

   A millimetric droplet may bounce and self-propel on the surface of a vertically vibrating bath, where its horizontal “walking” motion is induced by repeated impacts with its accompanying Faraday wave field. For ergodic long-time dynamics, we derive the relationship between the droplet’s stationary statistical distribution and its mean wave field in a very general setting. We then focus on the case of a droplet subjected to a harmonic potential with its motion confined to a line. By analyzing the system’s periodic states, we reveal a number of dynamical regimes, including those characterized by stationary bouncing droplets trapped by the harmonic potential, periodic quantized oscillations, chaotic motion and wavelike statistics, and periodic wave-trapped droplet motion that may persist even in the absence of a central force. We demonstrate that as the vibrational forcing is increased progressively, the periodic oscillations become chaotic via the Ruelle-Takens-Newhouse route. We rationalize the role of the local pilot-wave structure on the resulting droplet motion, which is akin to a random walk. We characterize the emergence of wavelike statistics influenced by the effective potential that is induced by the mean Faraday wave field. 

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4. The Stability of a Hydrodynamic Bravais Lattice

   https://doi.org/10.3390/sym14081524  

   Couchman, Miles M.; Evans, Davis J.; Bush, John W. (August 2022, Symmetry)

   We present the results of a theoretical investigation of the stability and collective vibrations of a two-dimensional hydrodynamic lattice comprised of millimetric droplets bouncing on the surface of a vibrating liquid bath. We derive the linearized equations of motion describing the dynamics of a generic Bravais lattice, as encompasses all possible tilings of parallelograms in an infinite plane-filling array. Focusing on square and triangular lattice geometries, we demonstrate that for relatively low driving accelerations of the bath, only a subset of inter-drop spacings exist for which stable lattices may be achieved. The range of stable spacings is prescribed by the structure of the underlying wavefield. As the driving acceleration is increased progressively, the initially stationary lattices destabilize into coherent oscillatory motion. Our analysis yields both the instability threshold and the wavevector and polarization of the most unstable vibrational mode. The non-Markovian nature of the droplet dynamics renders the stability analysis of the hydrodynamic lattice more rich and subtle than that of its solid state counterpart. 

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5. The energetics of pilot-wave hydrodynamics

   https://doi.org/10.1017/jfm.2025.168  

   Durey, Matthew; Bush, John WM (April 2025, Journal of Fluid Mechanics)

   A millimetric droplet may bounce and self-propel across the surface of a vertically vibrating liquid bath, guided by the slope of its accompanying Faraday wave field. The ‘walker’, consisting of a droplet dressed in a quasi-monochromatic wave form, is a spatially extended object that exhibits many phenomena previously thought exclusive to the quantum realm. While the walker dynamics can be remarkably complex, steady and periodic states arise in which the energy added by the bath vibration necessarily balances that dissipated by viscous effects. The system energetics may then be characterised in terms of the exchange between the bouncing droplet and its guiding or ‘pilot’ wave. We here characterise this energy exchange by means of a theoretical investigation into the dynamics of the pilot-wave system when time-averaged over one bouncing period. Specifically, we derive simple formulae characterising the dependence of the droplet’s gravitational potential energy and wave energy on the droplet speed. Doing so makes clear the partitioning between the gravitational, wave and kinetic energies of walking droplets in a number of steady, periodic and statistically steady dynamical states. We demonstrate that this partitioning depends exclusively on the ratio of the droplet speed to its speed limit. 

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