MATHEMATICAL-ANALYSIS OF SIDE-BAND INSTABILITIES WITH APPLICATION TO RAYLEIGH-BENARD CONVECTION

We introduce a new method for the analysis of sideband instabilities w hich are important for periodic patterns appearing in systems close to the instability threshold. The method relies on a two-fold applicatio n of the Liapunov-Schmidt reduction procedure, a first application to the nonlinear bifurcation problem and a second application to the line ar spectral problem. We obtain rigorous results on the spectrum of the associated linearization in spaces allowing for general sideband pert urbations by treating the sideband vector and the spectral parameter a s small bifurcation parameters. We apply the theory to the small roll solutions in the Rayleigh-Benard convection and derive domains in Rayl eigh, Prandtl, and wave number space where the rolls are unstable. We recover the Eckhaus, zigzag, and skew-varicose instabilities obtained earlier by formal methods.