= Stationary front of a cubic-quintic amplitude equation
{title2=$\mu_M=-3\alpha^2/16$}
A nontrivial stationary front of the <real-coefficient cubic-quintic amplitude equation> joining a nonzero pattern to zero must have constant phase. Writing $A=Re^{i\phi}$ gives $R^2\phi'=J$; 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 $A=e^{i\phi_0}\sqrt{(3\alpha/4)/(1+e^{\sqrt3\alpha(x-x_0)/2})}$. This proves existence as well as the necessary equal-potential condition. The zero solution is excluded from the term front.
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