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

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.