Mathematical Analysis of Sideband Instabilities with Application to Rayleigh{b Enard Convection

We introduce a new method for the analysis of sideband instabilities which are important for periodic patterns appearing in systems close to the instability threshold. The method relies on a two{fold application of the Liapunov{Schmidt reduction procedure, a rst application to the nonlinear bifurcation problem and a second application to the linear spectral problem. We obtain rigorous result on the spectrum of the associated linearization in spaces allowing for general sideband perturbations by treating the sideband vector and the spectral parameter as small bifurcation parameters. We apply the theory to the small roll solutions in Rayleigh{B enard convection and derive domains in Rayleigh, Prandtl and wave number space where the rolls are unstable. We recover the Eckhaus, zigzag, and the skew{varicose instabilities obtained earlier by formal methods.