  In the classical electrodynamics of point charges in vacuum, the
electromagnetic field, and therefore the Lorentz force, is ill-defined at the
locations of the charges. Kiessling resolved this problem by using the momentum
balance between the field and the particles, extracting an equation for the
force that is well-defined where the charges are located, so long as the field
momentum density is locally integrable in a neighborhood of the charges.
  In this paper, we examine the effects of such a force by analyzing a
simplified model in one space dimension. We study the joint evolution of a
massless scalar field together with its singularity, which we identify with the
trajectory of a particle. The static solution arises in the presence of no
incoming radiation, in which case the particle remains at rest forever. We will
prove the stability of the static solution for particles with positive bare
mass by showing that a pulse of incoming radiation that is compactly supported
away from the point charge will result in the particle eventually coming back
to rest. We will also prove the nonlinear instability of the static solution
for particles with negative bare mass by showing that an incoming radiation
with arbitrarily small amplitude will cause the particle to reach the speed of
light in finite time. We conclude by discussing modifications to this simple
model that could make it more realistic.
