Derivation of the amplitude equation at the Rayleigh–Bènard instability

Abstract

The full amplitude equation (as introduced by Newell and Whitehead and Segel) is derived directly from the hydrodynamic equations for both ’’free’’ and ’’rigid’’ upper and lower boundaries. The coefficients involved, including the interaction parameter of nonparallel rolls, are explicitly calculated for both boundary conditions and all Prandtl numbers.