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.

For a real Ginzburg–Landau equation plane wave with amplitude squared $b/g>0$, perturbing its amplitude and phase couples Fourier modes on either side of its carrier wavenumber. The displayed eigenvalues follow from the real-linear two-component perturbation equation. Their small-$p$ phase branch is $-\xi (\mu -3\xi q^2)p^2/(\mu -\xi q^2)+O(p^4)$, so the robust infinite-domain stable band is $q^2<\mu /(3\xi )$. On a finite periodic envelope domain only discrete $p$ are allowed; an instability requires an allowed $p$ with $p^2<6q^2-2\mu /\xi$. Uniform amplitude perturbations alone cannot determine this Eckhaus instability.

For the upper plane-wave branch of the quintic real Ginzburg-Landau equation, put $\Lambda =2R^2(2R^2-\alpha )>0$. A modulation of wavenumber $p$ has growth rates $-p^2-\Lambda /2\pm \sqrt{{\Lambda }^2/4+4Q^2{p}^2}$. Long-wave phase diffusion is positive when $4Q^2<\Lambda$, giving the displayed stability boundary. At equality the leading phase growth is $-p^4/\Lambda$, so the boundary is marginal in the diffusion coefficient but damped at the next spatial order. In the $(\mu ,Q)$ plane the boundary is $\mu =2Q^2-\alpha [\alpha +\sqrt{{\alpha }^2+16Q^2}]/8$.