 Solution

Let the uniform nonzero amplitude be $R$, so $\mu =R^2-\alpha R$ and $B=0$. Write $r=R(2R-\alpha )$. A perturbation with Fourier wavenumber $k=n\pi /L$ has stability matrixThe constant $B$ mode is disallowed because the conserved mean is prescribed. At $k=0$ the only allowed mode has eigenvalue $-r$, so homogeneous stability requires $r>0$. Then every nonzero-mode trace is negative, and its determinant is smallest in sign at the smallest allowed $k^2$. The exact periodic stability criterion for a conserved-field uniform amplitude isA zero in either inequality is a marginal case requiring nonlinear analysis. The zero-amplitude branch, separately, has amplitude growth rates $\mu -k^2$ and nonzero conserved-field growth rates $-\sigma k^2$, and is strictly stable when $\mu <0$.

For the long-domain interpretation one may take $\alpha >0$ by reversing the sign of $A$, and first consider the positive upper branch $R>\alpha /2$. It is homogeneously stable even though sufficiently long spatial modulations grow wheneverFor $0<\delta <1$, this givesAt a fixed finite $L$ the actual unstable part is smaller: it must also satisfy $(\pi /L{)}^2<\alpha R-2(1-\delta )R^2$. For $\delta \ge 1$, the right side is positive for every positive upper-branch $R$, so every such state is unstable to sufficiently long waves. The physical mechanism is feedback from the conserved field: an increase in $A^2$ reduces local $B$, and the term $-AB$ reinforces that increase.

There are two important limits on the wording of the requested stability conclusion. First,never vanishes at a finite regular point of this branch. At its fold the derivative is infinite. Thus a condition of zero $dR/d\mu$ cannot literally occur here. The meaningful contrast is that a positive-slope, homogeneously stable branch can still have a growing spatial mode.

Second, a long-wave conclusion cannot be asserted for every finite periodic domain. For example, $\alpha =1$, $\delta =2$, $\sigma =1$, $R=1$, $\mu =0$, $L=1$ give $r=1$ and $k^2+r-2\delta R^2=n^2{\pi }^2-3>0$ for every nonzero integer $n$. This state is strictly stable even though $\delta >1$. Nor does the unrestricted statement include every sign of $R$: $\alpha =1$, $\delta =2$, $R=-1/10$, $\mu =11/100$ give $r=3/25$ and $r-2\delta R^2=2/25>0$, hence stability for every period and every $\sigma >0$. These examples distinguish the exact stated real-variable periodic problem from its intended positive-amplitude, long-layer interpretation.