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For the real-coefficient cubic-quintic amplitude equation, the trivial pattern $A=0$ exists for every $\mu$. A nonzero spatially uniform pattern has arbitrary constant phase and intensity $q=|A{|}^2>0$ satisfying $\mu +\alpha q-q^2=0$. ThusNonzero patterns exist for $\mu \ge -{\alpha }^2/4$. The upper branch $q_{+}$ exists throughout that range; a distinct positive lower branch $q_{-}$ exists only for $-{\alpha }^2/4<\mu <0$ and reaches zero at $\mu =0$.

For a homogeneous perturbation, write $A=R{e}^{i\varphi }$. Then $\dot{R}=R(\mu +\alpha R^2-R^4)$ and $\dot{\varphi }=0$. The radial eigenvalue at a nonzero pattern iswhile the phase eigenvalue is zero. Therefore the upper branch is stable modulo its arbitrary phase for $\mu >-{\alpha }^2/4$; the lower branch is unstable. The phase-neutral direction prevents asymptotic attraction to one particular complex amplitude, but permits orbital stability of the circle of equivalent patterns.

Conduction is asymptotically stable for $\mu <0$ and unstable for $\mu >0$. At $\mu =0$, the positive cubic term destabilizes arbitrarily small nonzero amplitudes, even though its linear eigenvalue vanishes. At $\mu =-{\alpha }^2/4$, the merged nonzero branch is semistable from above only: $\dot{R}=-R(R^2-\alpha /2{)}^2$, so amplitudes just below it eventually decrease toward zero. Thus the bistable interval is $-{\alpha }^2/4<\mu <0$, with the endpoints treated separately rather than declared stable merely because an eigenvalue is zero.

For the intended nontrivial interface, take the limiting amplitude at the patterned end to be $R_{-}>0$; the identically zero solution is otherwise a trivial exception. Splitting the stationary real-coefficient cubic-quintic amplitude equation into real and imaginary parts givesThe imaginary equation implies $R^2{\varphi }^{\prime }=J$, a constant. The radial first integral isA regular solution has finite $E$. As $R\to 0$, the nonnegative $J^2/(2R^2)$ term would diverge unless $J=0$. Hence ${\varphi }^{\prime }=0$ wherever the nontrivial front has positive amplitude. This conclusion does not assume in advance that the patterned end has no phase gradient.

For a heteroclinic orbit connecting two constant states, $R^{\prime }$ tends to zero at both ends. Thus $E=V(0)=0$, while the nonzero end must obey both $\mu +\alpha q-q^2=0$ and $V(\sqrt{q})=0$. Eliminating $\mu$ givesThis is the stationary front of a cubic-quintic amplitude equation: a unique equal-potential parameter in the bistable interval.

Existence, not just a necessary condition, follows from the first integral. At ${\mu }_M$, $V(R)=-R^2(R^2-q{)}^2/6$. The descending solution obeys $R^{\prime }=-R(q-R^2)/\sqrt{3}$, which integrates toIt connects the stable upper pattern at $x\to -\infty$ to conduction at $x\to +\infty$, with arbitrary translation $x_0$ and constant phase ${\varphi }_0$.

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.