Title: Stability of spatially periodic pulse patterns in a class of singularly perturbed reaction-diffusion equations

Authors: Arjen Doelman and Harmen van der Ploeg

Issue: Volume 54 (2005), Issue 5, 1219-1302

Abstract: In this paper we develop a stability theory for spatially periodic patterns on $\mathbb{R}$. Our approach is valid for a class of singularly perturbed reaction-diffusion equations that can be represented by the generalized Gierer-Meinhardt equations as 'normal form'. These equations exhibit a large variety of spatially periodic patterns. We construct an Evans function $\mathcal{D}(\lambda,\gamma)$ that is defined for the $\gamma$-eigenvalue $\lambda$ in a certain subset of $\mathbb{C}$. The spectrum associated to the stability of the periodic pattern is given by the solutions $\lambda(\gamma)$ of $\mathcal{D}(\lambda(\gamma),\gamma) = 0$, where $\gamma \in \mathbf{S}^1$. Although our method can be applied to all types of singular pulse patterns, we focus on the stability analysis of the families of most simple periodic solutions. By decomposing $\mathcal{D}(\lambda,\gamma)$ into a product of a 'slow' and a 'fast' Evans function, we are able to determine explicit expressions for the $\gamma$-eigenvalues that are $\mathcal{O}(1)$ with respect to the small parameter $\epsilon$. Although the branch of 'small' $\gamma$-eigenvalues that is connected to the translational $1$-eigenvalue $\lambda(1) = 0$ cannot be studied by this decomposition, our methods also enable us determine the location of these $\gamma$-eigenvalues. Thus, our approach provides a full analytical control of the (spectral) stability of the singular spatially periodic patterns. We establish that the destabilization of a periodic pulse pattern on $\mathbb{R}$ is always initiated by the $\mathcal{O}(1)$ $\gamma$-eigenvalues, and consider various kinds of bifurcations. Finally, we apply our insights to the stability problem associated to the restriction of a periodic pulse pattern to a bounded domain with homogeneous Neumann boundary conditions.