The Stability and Evolution of Curved Domains Arising from One-Dimensional Localized Patterns

Abstract

In many pattern forming systems, narrow two-dimensional domains can arise whose cross sections are roughly one-dimensional localized solutions. This paper investigates this phenomenon in the variational Swift-Hohenberg equation. Stability of straight line solutions is analyzed, leading to criteria for either curve buckling or curve disintegration. Matched asymptotic expansions are used to derive a two-term expression for the geometric motion of curved domains, which includes both elastic and surface diffusion-type regularizations of curve motion. This leads to novel equilibrium curves and space-filling pattern proliferation. Numerical tests are used to confirm and illustrate these phenomena.

A numerical framework for singular limits of a class of reaction diffusion problems

“…Up to this point, all of the instability patterns have stabilized but as a final demonstration we consider curve expansion, also known as curve buckling (cf. [23], [13]). Consider taking a circular domain R = 10 where inside we initialize a curve Γ as Γ :{(r, θ)|r = 5 + 0.02 cos(2θ) + 0.3 cos(3θ)+ 0.04 cos(6θ) + 0.08 cos(12θ), θ ∈ [0, 2π]}.…”

“…In order to avoid this logarithmic singularity in the integral, we will add and subtract a logarithmic term from (13). However, due to the periodicity of the curve, if σ * = 0 or σ * is near 1 there are singularity effects from periodic extensions of the curve and as such we will also remove image singularities one full period away from σ * on either side.…”

“…Since then, reaction diffusion models have been studied extensively in the context of pattern formation problems (cf. [40], [26], [13], [36]) whereby coherent structures emerge from some general (often random) background initial data. Such localized patterns have been observed in varied systems such as animal spotting [31], sea-shell formation [27], urban crime analysis [25], and animal aggregation [7].…”

Existence, Stability, and Dynamics of Ring and Near-Ring Solutions to the Saturated Gierer--Meinhardt Model in the Semistrong Regime

“…[11], [13]) (x, y) → (η, s) where η is the signed normal distance from a curve (inward is positive) and s is the curve arclength. In this coordinate frame, and sufficiently close to the curve,…”

“…[32]), the Swift-Hohenberg model (cf. [13]), a generalized Schnakenberg system modelling root hair initiation in plants (cf. [5]), a model for urban crime (cf.…”