 Real-coefficient cubic-quintic amplitude equation

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 $e^{i{\varphi }_0}$ 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.