  Plasticity of metals is the emergent phenomenon of many crystal defects
(dislocations) which interact and move on microscopic time and length scales.
Two of the commonly used models to describe such dislocation dynamics are the
Peierls-Nabarro model and the so-called discrete dislocation dynamics model.
  However, the consistency between these two models is known only for a few
number of dislocations or up to the first time at which two dislocations
collide. In this paper we resolve these restrictions, and establish the
consistency for any number of dislocations and without any restriction on their
initial position or orientation.
  In more detail, the evolutive Peierls-Nabarro model which we consider
describes the evolution of a phase-field function $v_\e(t,x)$ which represents
the atom deformation in a crystal. The model is a reaction-diffusion equation
of Allen-Cahn type with the half Laplacian. The small parameter $\ep$ is the
ratio between the atomic distance and the typical distance between phase
transitions in $v_\e$. The position of a phase transition determines the
position of a dislocation, and the sign of the transition (up or down)
determines the orientation.
  The goal of this paper is to derive the asymptotic behavior of the function
$v_\e$ as $\ep\to0$ up to arbitrary end time $T$; in particular beyond
collisions. We prove that $v_\e$ converges to a piecewise constant function
$v$, whose jump points in the spatial variable satisfy the ODE system which
represents discrete dislocation dynamics with annihilation. Our proof method is
to explicitly construct and patch together several sub- and supersolutions of
$v_\e$, and to show that they converge to the same limit $v$.
