The discrete dislocation dynamics of multiple dislocation loops

We consider a nonlocal reaction-diffusion equation that physically arises from the classical Peierls-Nabarro model for dislocations in crystalline structures. Our initial configuration corresponds to multiple slip loop dislocations in ${\mathbb{R}}^n$, $n\ge 2$. After suitably rescaling the equation with a small phase parameter $\epsilon >0$, the rescaled solution solves a fractional Allen-Cahn equation. We show that, as $\epsilon \to 0$, the limiting solution exhibits multiple interfaces evolving independently and according to their mean curvature.