Instability and localisation of patterns due to a conserved quantity

We describe the influence of a conserved quantity on the stability properties of roll, square and hexagonal patterns near stationary onset. The appropriate systems of amplitude equations are analysed: they comprise equations for the amplitudes of the pattern modes, together with an additional coupled equation governing the evolution of a large-scale mode associated with the conserved quantity. We find that the presence of this large-scale mode may result in the destabilisation of all regular roll, square or hexagonal patterns, leading to amplitude modulation and strong localisation of the pattern. Our analytical results are complemented throughout by numerical simulations of a model partial differential equation, to illustrate the modulational instabilities and their nonlinear development.