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Destabilization of Fronts in a Class of Bistable Systems

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Title: Destabilization of Fronts in a Class of Bistable Systems
Authors: Doelman, Arjen Browse this author
Iron, David Browse this author
Nishiura, Yasumasa Browse this author →KAKEN DB
Keywords: pattern formation
bistable systems
geometric singular perturbation theory
stability analysis
Evans functions
AMS subject classifications = 35B25
AMS subject classifications = 35B32
AMS subject classifications = 35B35
AMS subject classifications = 35K57
AMS subject classifications = 35P20
AMS subject classifications = 34A26
AMS subject classifications = 34C37
Issue Date: 2004
Publisher: Society for Industrial and Applied Mathematics
Journal Title: SIAM Journal on Mathematical Analysis
Volume: 35
Issue: 6
Start Page: 1420
End Page: 1450
Publisher DOI: 10.1137/S0036141002419242
Abstract: In this article, we consider a class of bistable reaction-diffusion equations in two components on the real line. We assume that the system is singularly perturbed, i.e., that the ratio of the diffusion coefficients is (asymptotically) small. This class admits front solutions that are asymptotically close to the (stable) front solution of the “trivial” scalar bistable limit system ut = uxx+u(1−u2). However, in the system these fronts can become unstable by varying parameters. This destabilization is caused by either the essential spectrum associated to the linearized stability problem or by an eigenvalue that exists near the essential spectrum. We use the Evans function to study the various bifurcation mechanisms and establish an explicit connection between the character of the destabilization and the possible appearance of saddle-node bifurcations of heteroclinic orbits in the existence problem.
Rights: © 2004 Society for Industrial and Applied Mathematics
Type: article
Appears in Collections:電子科学研究所 (Research Institute for Electronic Science) > 雑誌発表論文等 (Peer-reviewed Journal Articles, etc)

Submitter: 西浦 廉政

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