Nonlinear diffusion of dislocation density and self-similar solutions

TL;DR

This paper investigates a nonlinear pseudodifferential equation modeling dislocation dynamics, revealing that solutions tend to self-similar profiles over long times, which enhances understanding of material deformation processes.

Contribution

It introduces a new analysis of the long-term behavior of solutions to a nonlinear dislocation equation, highlighting self-similarity in the asymptotics.

Findings

Solutions exhibit self-similar asymptotic profiles

Long-time behavior characterized by specific scaling laws

Provides mathematical insight into dislocation evolution

Abstract

We study a nonlinear pseudodifferential equation describing the dynamics of dislocations. The long time asymptotics of solutions is described by the self-similar profiles.