Spectra and Stability of Spatially Periodic Pulse Patterns II: The Critical Spectral Curve

Abstract

In the stability analysis of pulse patterns in singularly perturbed reaction-diffusion systems, the scale separation is often exploited to reduce complexity. There are various methods to decompose the spectrum of the linearization about the pattern into slow and fast pieces and thus gain control over the spectrum in the asymptotic limit. These reduction methods have been developed in prototype models such as the FitzHugh--Nagumo, Gray--Scott, and Gierer--Meinhardt equations, which are of slowly linear nature; i.e., the slow dynamics away from the pulses are driven by linear equations. Recently, these methods were extended to homoclinic pulses in a general, slowly nonlinear class of reaction-diffusion systems. Yet, a straightforward extension to periodic pulse solutions is impossible due to two major obstructions. First, the reduction methods that lead to asymptotic spectral control for the homoclinic pulses fail for periodic patterns in slowly nonlinear systems. Second, there is a curve of spectrum attached to the origin that shrinks to the origin in the asymptotic limit, whereas for homoclinic pulses only a simple eigenvalue resides at the origin. Therefore, controlling the spectrum in the asymptotic limit is insufficient to decide upon stability. The first obstruction has been addressed in the companion paper [B. de Rijk, A. Doelman, and J. D. M. Rademacher, SIAM J. Math. Anal., 48 (2016), pp. 61--121]. In this paper we focus on the second obstruction: we obtain leading-order control over the critical spectral curve attached to the origin. The proof relies on Lin's method and utilizes exponential trichotomies. Our results yield explicit conditions for nonlinear stability and instability for stationary, spatially symmetric, periodic pulses in a large class of multicomponent, slowly nonlinear, reaction-diffusion systems. Moreover, we gain insight into the associated destabilization mechanisms.