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

Abstract

In this paper we develop a stability theory for spa- tially periodic patterns onR. Our approach is valid for a class of singularly perturbed reaction-diVusion equations that can be rep- resented by the generalized Gierer-Meinhardt equations as 'nor- mal form'. These equations exhibit a large variety of spatially pe- riodic patterns. We construct an Evans function DN;O that is defined for the-eigenvalue in a certain subset ofC. The spec- trum associated to the stability of the periodic pattern is given by the solutionsNO ofDNNO;OE 0, where 2 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 decomposingDN;O into a product of a 'slow' and a 'fast' Evans function, we are able to determine explicit expressions for the-eigenvalues that areON1O with respect to the small parameter. Although the branch of 'small'-eigenvalues that is connected to the translational 1-eigenvalueN1OE 0c an- not be studied by this decomposition, our methods also enable us determine the location of these-eigenvalues. Thus, our ap- proach 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 onR is always initiated by theON1O-eigenvalues, and consider various kinds of bifurca- tions. Finally, we apply our insights to the stability problem asso- ciated to the restriction of a periodic pulse pattern to a bounded domain with homogeneous Neumann boundary conditions.