Instability and Stability of Rolls in the Swift–Hohenberg Equation

We develop a method for the stability analysis of bifurcating spatially periodic patterns under general nonperiodic perturbations. In particular, it enables us to detect sideband instabilities. We treat in all detail the stability question of roll solutions in the two–dimensional Swift–Hohenberg equation and derive a condition on the amplitude and the wave number of the rolls which is necessary and sufficent for stability. Moreover, we characterize the set of those wave vectors which give rise to unstable perturbations. Dedicated to Professor K. Kirchg?ssner on the occasion of his sixty-fifth birthday Received: 25 October 1996 / Accepted: 24 March 1997