A variational theory for point defects in patterns

TL;DR

This paper develops a rigorous variational framework for understanding point defects in pattern-forming systems, aligning theoretical predictions with experimental and numerical observations.

Contribution

It introduces a variational theory that rigorously describes point defects in pattern formation models, bridging microscopic simulations and experimental findings.

Findings

Derived a scaling law for energy-minimizing solutions with defects

Showed that defects in the model match those observed experimentally

Validated the theory with numerical simulations

Abstract

We derive a rigorous scaling law for minimizers in a natural version of the regularized Cross-Newell model for pattern formation far from threshold. These energy-minimizing solutions support defects having the same character as what is seen in experimental studies of the corresponding physical systems and in numerical simulations of the microscopic equations that describe these systems.