  A classical wave-particle entity (WPE) can be realized experimentally as a
droplet walking on the free surface of a vertically vibrating liquid bath, with
the droplet's horizontal walking motion guided by its self-generated wave
field. These self-propelled WPEs have been shown to exhibit analogs of several
quantum and optical phenomena. Using an idealized theoretical model that takes
the form of a Lorenz-like system, we theoretically and numerically explore the
dynamics of such a one-dimensional WPE in a sinusoidal potential. We find
steady states of the system that correspond to a stationary WPE as well as a
rich array of unsteady motions such as back-and-forth oscillating walkers,
runaway oscillating walkers and various types of irregular walkers. In the
parameter space formed by the dimensionless parameters of the applied
sinusoidal potential, we observe patterns of alternating unsteady behaviors
suggesting interference effects. Additionally, in certain regions of the
parameter space, we also identify multistability in the particle's long-term
behavior that depends on the initial conditions. We make analogies between the
identified behaviors in the WPE system and Bragg's reflection of light as well
as electron motion in crystals.
