= Real-coefficient cubic-quintic amplitude equation
{title2=$A_t=\mu A+\alpha|A|^2A-|A|^4A+A_{xx}$}
For real $\mu$ and $\alpha>0$, this complex <amplitude equation> has a destabilizing cubic term and a saturating quintic term. Nonzero uniform intensities solve $\mu+\alpha q-q^2=0$. The upper branch is radially stable and the lower branch radially unstable; constant phase is neutral because multiplication by $e^{i\phi_0}$ preserves the equation. Stable conduction and the stable upper pattern coexist for $-\alpha^2/4<\mu<0$. At the fold, a zero radial eigenvalue conceals one-sided nonlinear instability, so strict stability cannot be inferred from eigenvalues alone.
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