Real-coefficient cubic-quintic amplitude equation ($A_t=\mu A+\alpha |A{|}^{2}A-|A{|}^{4}A+A_{xx}$)

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

A nontrivial stationary front of the real-coefficient cubic-quintic amplitude equation joining a nonzero pattern to zero must have constant phase. Writing $A=R{e}^{i\varphi }$ gives $R^2{\varphi }^{\prime }=J$; the finite first integral contains $J^2/(2R^2)$, so approaching zero forces $J=0$. Equal endpoint potentials then give $R_{-}^2=3\alpha /4$ and the unique parameter ${\mu }_M=-3{\alpha }^2/16$. The explicit profile is $A=e^{i{\varphi }_0}\sqrt{(3\alpha /4)/(1+e^{\sqrt{3}\alpha (x-x_0)/2})}$. This proves existence as well as the necessary equal-potential condition. The zero solution is excluded from the term front.