The concertina pattern

Abstract

This is a continuation of a series of papers on the concertina pattern. The concertina pattern is a ubiquitous metastable, nearly periodic magnetization pattern in elongated thin film elements. In previous papers, a reduced variational model for this pattern was rigorously derived from 3-d micromagnetics. Numerical simulations of the reduced model reproduce the concertina pattern and show that its optimal period $${\widehat{w}_{opt}}$$ is an increasing function of the applied external field $${\widehat{h}_{ext}}$$ . The latter is an explanation of the experimentally observed coarsening. Domain theory, which can be heuristically derived from the reduced model, predicts and quantifies this dependence of $${\widehat{w}_{opt}}$$ on $${\widehat{h}_{ext}}$$ . In this paper, we rigorously extract these heuristic observations of domain theory directly from the reduced model. The main ingredient of the analysis is a new type of estimate on solutions of a perturbed Burgers equation.