= Long-wave instability with a conserved mean field
Putting $\lambda=l^2\nu$ into the sideband cubic and retaining $l^4$ gives
$$
r\nu^2-[(\mu-\sigma-1)r+2q^2]\nu-[(\mu-\sigma)r+2\sigma q^2]=0.
$$
For $r=1-q^2>0$, $\mu/\sigma>(1-3q^2)/(1-q^2)$ makes the constant coefficient negative. There is a positive real $\nu$, so sufficiently small nonzero $l$ grows. On a finite period $L$, however, only $l=2\pi j/L$ are allowed. For $q=0$, $\mu=2$, $\sigma=1$, $L=1$, every nonzero allowed mode has negative growth rates despite this long-wave inequality. Thus an arbitrarily-small-wave-number conclusion cannot be asserted without the domain-size qualification.
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