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Homoclinic stripe patterns. (English) Zbl 1004.35063

Summary: We study homoclinic stripe patterns in the two-dimensional generalized Gierer-Meinhardt equation, where we interpret this equation as a prototypical representative of a class of singularly perturbed monostable reaction-diffusion equations. The structure of a stripe pattern is essentially one-dimensional; therefore, we can use results from the literature to establish the existence of the homoclinic patterns. However, we extend these results to a maximal domain in the parameter space and establish the existence of a bifurcation that forms a new upper bound on this domain. Beyond this bifurcation, the Gierer-Meinhardt equation exhibits self-replicating pulse, respectively, stripe patterns in one, respectively, two dimension(s). The structure of the self-replication process is very similar to that in the Gray-Scott equation.
We investigate the stability of the homoclinic stripe patterns by an Evans function analysis of the associated linear eigenvalue problem. We extend the recently developed nonlocal eigenvalue problem (NLEP) approach to two-dimensional systems. Except for a region near the upper bound of the domain of existence in parameter space, this method enables us to get explicit information on the spectrum of the linear problem. We prove that, in this subregion, all homoclinic stripe patterns must be unstable as solutions on \(\mathbb{R}^2\). However, stripe patterns can be stable on domains of the type \(\mathbb{R} \times (0,L_y)\). Our analysis enables us to determine an upper bound on \(L_y\); moreover, the analysis indicates that stripe patterns can become stable on \(\mathbb{R}^2\) near the upper bound of the existence domain. This is confirmed numerically: It is shown by careful simulations that there can be stable homoclinic stripe patterns on \(\mathbb{R}^2\) for parameter values near the self-replication bifurcation.

MSC:

35K57 Reaction-diffusion equations
35B25 Singular perturbations in context of PDEs
35B32 Bifurcations in context of PDEs
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
92C15 Developmental biology, pattern formation
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