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 $\mathbb R^n$, $n\geq 2$. 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.

This is a joint work with Mary Vaughan  (University of Western Australia).