= Sideband spectrum of a real Ginzburg-Landau plane wave
{title2=$\sigma_\pm=-\xi p^2-b\pm\sqrt{b^2+4\xi^2q^2p^2},\quad b=\mu-\xi q^2$}
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>.
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