A fractional glance to the theory of edge dislocations

TL;DR

This paper explores advanced mathematical models of edge dislocations in crystals using fractional Laplace equations, analyzing patterns, dynamics, and long-term behavior of dislocation solutions.

Contribution

It introduces new insights into dislocation theory by applying fractional Laplace models to analyze complex dislocation patterns and their dynamics.

Findings

Existence of heteroclinic, homoclinic, and multibump dislocation patterns

Analysis of large-scale behavior of dislocation solutions

Insights into dislocation dynamics and asymptotics after collisions

Abstract

We revisit some recents results inspired by the Peierls-Nabarro model on edge dislocations for crystals which rely on the fractional Laplace representation of the corresponding equation. In particular, we discuss results related to heteroclinic, homoclinic and multibump patterns for the atom dislocation function, the large space and time scale of the solutions of the parabolic problem, the dynamics of the dislocation points and the large time asymptotics after possible dislocation collisions.