Essential instabilities of fronts: bifurcation, and bifurcation failure

Various instability mechanisms of fronts in reaction-diffusion systems are analysed: the emphasis is on instabilities that have the potential to creat e modulated (i.e. time-periodic) waves near the primary front. Hopf bifurca tions caused by point spectrum with associated localized eigenfunctions pro vide an example of such an instability. A different kind of instability occ urs if one of the asymptotic rest states destabilizes: these instabilities are caused by essential spectrum. It is demonstrated that. if the rest stat e ahead of the Front destabilizes, then modulated fronts are created that c onnect the rest state behind the front with small spatially periodic patter ns ahead of the front. These modulated fronts are stable provided the spati ally periodic patterns are stable. IL on the other hand, the rest state beh ind the front destabilizes, then modulated fronts that leave a spatially pe riodic pattern behind do not exist.