A Rigorous Floquet Approach to Parabolic PDEs, with Applications to Walking Droplet Statistics

Abstract: Floquet theory provides a powerful tool for studying time-periodic dynamics, allowing the stability and long-time behavior of any periodic, linear ODE to be recast in terms of a linear time-invariant system. However, except when dealing with Hilbert spaces or certain well-behaved classes of operators, Floquet theory cannot be rigorously applied to PDEs without dimensional reduction. We describe a simple, rigorous extension of Floquet theory to a wide class of parabolic PDEs on Banach spaces, enabling direct application with minimal changes from the classical theory. To demonstrate the utility of the extended Floquet formalism, we investigate the statistics of 'walking droplets': millimetric oil droplets that self-propel over the surface of a vibrating oil bath. Walking droplets exhibit striking, emergent statistical patterns that are not yet fully understood. For instance, a widespread correspondence has been observed between the statistics of the droplet and the time-average of its guiding wave, but it has only been proved rigorously for a single droplet in an unbounded domain of uniform depth (Durey, Milewski, and Bush, 2018). We apply our Floquet results to derive the droplet-wave statistical correspondence in a highly general case, allowing for arbitrary geometries, multiple droplet interactions, and non-resonant droplet bouncing. Beyond walking droplets, our framework provides a versatile tool for analyzing periodic PDEs more broadly, with promising applications to tidal flows and nutrient-phytoplankton-zooplankton (NPZ) models under daily cycles.