Can large inhomogeneities generate target patterns?

TL;DR

This paper demonstrates that large inhomogeneities with slow decay can generate target patterns in oscillatory media, extending previous results to include defects with algebraic decay.

Contribution

It introduces a new analysis for target pattern formation caused by large, algebraically decaying defects, using weighted Sobolev spaces and matched asymptotic approximations.

Findings

Large defects can generate target patterns.

The pattern frequency is small beyond all orders of the defect strength.

Existence of solutions is proved using the implicit function theorem.

Abstract

We study the existence of target patterns in oscillatory media with weak local coupling and in the presence of an impurity, or defect. We model these systems using a viscous eikonal equation posed on the plane, and represent the defect as a perturbation. In contrast to previous results we consider large defects, which we describe using a function with slow algebraic decay, i.e., $g \sim {\mathcal O}(1/|x|^m)$ for $m \in (1,2]$. We prove that these defects are able to generate target patterns and that, just as in the case of strongly localized impurities, their frequency is small beyond all orders of the small parameter describing their strength. Our analysis consists of finding two approximations to target pattern solutions, one which is valid at intermediate scales and a second one which is valid in the far field. This is done using weighted Sobolev spaces, which allow us to recover Fredholm properties of the relevant linear operators, as well as the implicit function theorem, which is then used to prove existence. By matching the intermediate and far field approximations we then determine the frequency of the pattern that is selected by the system.