= Solution
Assume $R^2=1-q^2>0$. The symbol $R$ in this part is the <convection roll> amplitude, not the <Rayleigh number> of the previous part. Set $r=R^2$ for the algebra. The <periodic boundary condition> requires $qL\in2\pi\mathbb Z$ and modulation <wavenumbers> $lL\in2\pi\mathbb Z$. The conserved-field equation preserves the imposed zero mean of $B$, since the integral of an $X$ derivative over a period vanishes.
Equating the $e^{ilX+\lambda T}$ coefficients after <linearization> of the <convection amplitude coupled to a conserved field> gives
$$
\begin{aligned}\lambda a_1&=-(l^2+2ql)a_1-r(a_1+a_2)-b,\\\lambda a_2&=-(l^2-2ql)a_2-r(a_1+a_2)-b,\\\lambda b&=-\sigma l^2b-\mu r l^2(a_1+a_2).\end{aligned}
$$
The conjugate sideband is needed because the variation of $|A|^2A$ couples a perturbation to its <complex conjugate>. Put $S=a_1+a_2$, $D=a_1-a_2$. The <linear operator> on $(S,D,b)$ is
$$
M_l=\begin{pmatrix}-l^2-2r&-2ql&-2\\-2ql&-l^2&0\\-\mu r l^2&0&-\sigma l^2\end{pmatrix}.
$$
Its <characteristic polynomial>, expressed conveniently without expanding every coefficient, is
$$
\boxed{(\lambda+\sigma l^2)\big[(\lambda+l^2)(\lambda+l^2+2r)-4q^2l^2\big]-2\mu r l^2(\lambda+l^2)=0.}
$$
At $l=0$ it has roots $-2r,0,0$. The phase root is neutral; the uniform $B$ root is excluded by the prescribed zero-flux-perturbation mean. Nonzero arbitrarily long-wave modes remain allowed on a sufficiently large domain. For fixed $r>0$, the two roots tending to zero have $\lambda=l^2\nu+O(l^4)$ at simple limiting roots. Dividing the polynomial by $2l^4$ yields
$$
\boxed{r\nu^2-[(\mu-\sigma-1)r+2q^2]\nu-[(\mu-\sigma)r+2\sigma q^2]=0.}
$$
Repeated limiting roots may require a further expansion, but do not affect the strict instability criterion. If $(\mu-\sigma)r+2\sigma q^2>0$, the constant term is negative and the two real roots have opposite signs. The positive root gives a growing long-wave <sideband instability>. Using $r=1-q^2$ gives
$$
\boxed{\frac\mu\sigma>\frac{1-3q^2}{1-q^2}.}
$$
This is sufficient for instability when a sufficiently small nonzero allowed $l$ is available. It is not an unconditional instability theorem for every fixed finite period. For example $q=0$, $\mu=2$, $\sigma=1$, $L=1$ satisfy the inequality, but the nonzero allowed $l$ obey $l^2\geq4\pi^2$. Their roots are $-l^2$ and $-l^2-1\pm\sqrt{1+4l^2}$, all negative. The prescribed zero mean removes the uniform flux root and the remaining uniform phase is merely neutral. The corrected conclusion is the <long-wave instability with a conserved mean field>, with domain size and mode availability stated.
Back to article page
