 Eckhaus instability

For $r,\xi ,g>0$, longitudinal phase modulation of a detuned convection roll yields diffusion coefficient $D_{\varphi }=\xi (r-3\xi Q^2)/(r-\xi Q^2)$. The expression follows by eliminating the fast amplitude correction from the coupled linear amplitude-phase equations at small modulation wavenumber. Thus convection rolls become unstable for $Q^2>r/(3\xi )$ although they exist for $Q^2<r/\xi$. At the boundary the leading phase diffusion vanishes and higher spatial orders matter.