Eckhaus instability
= Eckhaus instability
{c}
For $r,\xi,g>0$, longitudinal phase modulation of a detuned <convection roll> yields diffusion coefficient $D_\phi=\xi(r-3\xi Q^2)/(r-\xi Q^2)$. 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.