 Modulation cutoff of a cubic-quintic uniform pattern

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.