  We consider a reaction-diffusion equation with a half-Laplacian. In the case
where the solution is independent on time, the model reduces to the
Peierls-Nabarro model describing dislocations as transition layers in a phase
field setting. We introduce a suitable rescaling of the evolution equation,
using a small parameter $\varepsilon$. As $\varepsilon$ goes to zero, we show
that the limit dynamics is characterized by a system of ODEs describing the
motion of particles with two-body interactions. The interaction forces are in
$1/x$ and correspond to the well-known interaction between dislocations.
