= Eckhaus boundary for a subcritical quintic amplitude equation
{c}
{title2=$Q^2\leq R^4-\alpha R^2/2$}
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 $-p^2-\Lambda/2\pm\sqrt{\Lambda^2/4+4Q^2p^2}$. 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 $\mu=2Q^2-\alpha[\alpha+\sqrt{\alpha^2+16Q^2}]/8$.
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