期刊
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
卷 38, 期 1, 页码 51-65出版社
WILEY-BLACKWELL
DOI: 10.1002/mma.3049
关键词
Ginzburg-Landau; inhomogeneity
资金
- National Science Foundation [DMS-0806614, DMS-1311740]
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1311740, 0806614] Funding Source: National Science Foundation
We study the effects of adding a local perturbation in a pattern-forming system, taking as an example the Ginzburg-Landau equation with a small localized inhomogeneity in two dimensions. Measuring the response through the linearization at a periodic pattern, one finds an unbounded linear operator that is not Fredholm due to continuous spectrum in typical translation invariant or weighted spaces. We show that Kondratiev spaces, which encode algebraic localization that increases with each derivative, provide an effective means to circumvent this difficulty. We establish Fredholm properties in such spaces and use the result to construct deformed periodic patterns using the Implicit Function Theorem. We find a logarithmic phase correction, which vanishes for a particular spatial shift only, which we interpret as a phase-selection mechanism through the inhomogeneity. Copyright (c) 2013 John Wiley & Sons, Ltd.
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