Destabilization of Fronts in a Class of Bistable Systems

• Published in: SIAM journal on mathematical analysis Vol. 35; no. 6; pp. 1420 - 1450
• Main Authors: Doelman, Arjen, Iron, David, Nishiura, Yasumasa
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2004
• Subjects: Eigenvalues; Exact sciences and technology; Mathematical analysis; Mathematics; Ordinary differential equations; Partial differential equations; Sciences and techniques of general use; Patterning; Heteroclinic orbit; Mathematical analysis; Singular perturbation; Bifurcation; Evans function; Eigenvalue problem; Stability analysis; Reaction diffusion equation; Bistable system
• ISSN: 0036-1410; 1095-7154
• DOI: 10.1137/S0036141002419242

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.