Instability of small stationary localized solutions to a class of reversible 1+1 PDEs

We study stability properties of small stationary localized and periodic solutions for a class of PDEs reversible with respect to a spatial variable x in the case where the related stationary equation undergoes the reversible Hopf bifurcation. Under general assumptions we prove instability of localized and stability of periodic solutions. Proofs use the normal form of the stationary equation and certain features of differential operators of the form lambda A - M with M being a linear differential operator. Two examples are considered.