Stationary front of a cubic-quintic amplitude equation (${\mu }_M=-3{\alpha }^2/16$)

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.