  We consider a system of charged particles moving on the real line driven by
electrostatic interactions. Since we consider charges of both signs, collisions
might occur in finite time. Upon collision, some of the colliding particles are
effectively removed from the system (annihilation). The two applications we
have in mind are vortices and dislocations in metals.
  In this paper we reach two goals. First, we develop a rigorous solution
concept for the interacting particle system with annihilation. The main
innovation here is to provide a careful management of the annihilation of
groups of more than two particles, and we show that the definition is
consistent by proving existence, uniqueness, and continuous dependence on
initial data. The proof relies on a detailed analysis of ODE trajectories close
to collision, and a reparametrization of vectors in terms of the moments of
their elements.
  Secondly, we pass to the many-particle limit (discrete-to-continuum), and
recover the expected limiting equation for the particle density. Due to the
singular interactions and the annihilation rule, standard proof techniques of
discrete-to-continuum limits do not apply. In particular, the framework of
measures seems unfit. Instead, we use the one-dimensional feature that both the
particle system and the limiting PDE can be characterized in terms of
Hamilton--Jacobi equations. While our proof follows a standard limit procedure
for such equations, the novelty with respect to existing results lies in
allowing for stronger singularities in the particle system by exploiting the
freedom of choice in the definition of viscosity solutions.
