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 \(\varepsilon >0\), the rescaled solution solves a fractional Allen–Cahn equation. We show that, as \(\varepsilon \rightarrow 0\), the limiting solution exhibits multiple interfaces evolving independently and according to their mean curvature.