= Modulation cutoff of a cubic-quintic uniform pattern
{title2=$L<\frac{8\pi\sqrt{10}}{3s}\quad(A_0\ne0)$}
For $A_T=\mu A+3\hat sA|A|^2-10A|A|^4+4A_{XX}$, a nonzero uniform real state with $B=A_0^2$ obeys $\mu=-3\hat sB+10B^2$. Real amplitude sidebands grow at $6\hat sB-40B^2-4\ell^2$, 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 $X=\varepsilon^2x$, 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.
Back to article page
