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 Rⁿ, n ≥ 2. After suitably rescaling the equation with a small phase parameter ε>0, the rescaled solution solves a fractional Allen-Cahn equation. We show that, as ε → 0, the limiting solution exhibits multiple interfaces evolving independently and according to their mean curvature.