Classical pilot-wave dynamics in three dimensions

Inspired by the hydrodynamic pilot-wave system and its successes as a basis for hydrodynamic quantum analogues, we present the results of a theoretical investigation of classical pilot-wave dynamics in three dimensions. We consider a particle emitting a spherically symmetric monochromatic wave, then moving in response to the local wave gradient. We demonstrate that, beyond a critical coupling constant, the stationary state destabilizes into a self-propelling state. We compute the quasi-monochromatic form of the accompanying pilot-wave field, and its dependence on the relevant control parameters, specifically the particle inertia and longevity of the pilot-wave field. The similarities between the dynamics arising in the three-dimensional system and its two-dimensional counterpart suggest that many of the quantum features captured in two dimensions will be robust to the transition to three dimensions. We discover a family of dynamical states peculiar to the three-dimensional geometry considered, including helical spin states in which the particle velocity has both angular and rectilinear components. These helical spin states are found to be quantized in both radius of curvature and pitch length and illustrate how classical pilot-wave dynamics may support a feature reminiscent of quantum spin. More exotic dynamical states are also explored and detailed.