Geometric Relations of Absolute and Essential Spectra of Wave Trains

Abstract

We analyze geometric relations of absolute and essential spectra for certain linear operators on the real line with periodic coefficients. These spectra correspond to accumulation sets of eigenvalues for increasing domain length under separated and periodic boundary conditions, respectively. The main result shows that critical isolated sets of essential spectra contain absolute spectra and yields an algorithm for its numerical computation. Linearizations of reaction diffusion systems in wave trains are used as an illustration, and we present a detailed numerical study of absolute and essential spectra for a wave train in the Schnakenberg model.