Quintic real Ginzburg-Landau equation ($A_T=\mu A+\alpha |A{|}^{2}A-|A{|}^{4}A+A_{XX},\alpha >0$)

A cubic destabilizing term and quintic saturation describe a subcritical stationary pattern bifurcation. A plane wave $A=R{e}^{iQX}$ has squared amplitude $x=R^2$ satisfying $\mu -Q^2+\alpha x-x^2=0$. The nonzero branches meet at $x=\alpha /2$, $\mu =Q^2-{\alpha }^2/4$. The lower branch is unstable to amplitude perturbations; the upper branch can also undergo an Eckhaus instability of its phase.

For $A_T=\mu A+3\hat{s}A|A{|}^2-10A|A{|}^4+4A_{XX}$, a nonzero uniform real state with $B=A_0^2$ obeys $\mu =-3\hat{s}B+10B^2$. Real amplitude sidebands grow at $6\hat{s}B-40B^2-4{\ell }^2$, while phase sidebands grow at $-4{\ell }^2$. The maximum amplitude growth coefficient is $9{\hat{s}}^2/40$, attained at $B=3\hat{s}/40$. Thus nonzero uniform states cannot have modulation bifurcations if the smallest allowed slow wavenumber exceeds $3\hat{s}/(4\sqrt{10})$. For a physical domain of length $L$ and $X={\epsilon }^{2}x$, this gives $L<8\pi \sqrt{10}/(3s)$. The zero state is an exception: it has sideband thresholds $\mu =4{\ell }^2$ for every positive allowed wavenumber, so an assertion including all uniform states would be false.