  A droplet bouncing on the surface of a vertically vibrating liquid bath can
walk horizontally, guided by the waves it generates on each impact. This
results in a self-propelled classical particle-wave entity. By using a
one-dimensional theoretical pilot-wave model with a generalized wave form, we
investigate the dynamics of this particle-wave entity. We employ different wave
forms to understand the role played by both wave oscillations and spatial wave
decay in the walking dynamics. In addition to steady walking, we uncover a rich
set of unsteady motions such as oscillating walking, self-trapped oscillations
and irregular walking. We explore the dynamical and statistical aspects of
irregular walking and connect the droplet dynamics with the Lorenz equations,
the Langevin equation, and deterministic diffusion.
