The stability and evolution of curved domains arising from one-dimensional localized patterns

Karl B Glasner, Alan E. Lindsay

Research output: Contribution to journalArticle

8 Citations (Scopus)

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.

Original language English (US) 650-673 24 SIAM Journal on Applied Dynamical Systems 12 2 https://doi.org/10.1137/120893008 Published - 2013

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Surface diffusion
Disintegration
Buckling
Curve
Swift-Hohenberg Equation
Surface Diffusion
Variational Equation
Matched Asymptotic Expansions
Motion
Two-dimensional Systems
Proliferation
Straight Line
Regularization
Cross section
Term

Keywords

• Curvature
• Geometric motion
• Localized states
• Matched asymptotics
• Surface diffusion

ASJC Scopus subject areas

• Analysis
• Modeling and Simulation

Cite this

In: SIAM Journal on Applied Dynamical Systems, Vol. 12, No. 2, 2013, p. 650-673.

Research output: Contribution to journalArticle

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AB - 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.

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