Conserved-field sideband dispersion relation

Linearize around $A=R{e}^{iqX}$ and write $r=R^2>0$. For sideband wavenumber $l$, the sum and difference of the two amplitude sidebands, together with the field amplitude $b$, evolve underExpanding $\det (\lambda I-M_l)$ givesAt $l=0$, the eigenvalues are $-2r,0,0$; the field's constant mode must be removed if its mean is prescribed. Nonzero long-wave field modes are still dynamically active.

Putting $\lambda =l^2\nu$ into the sideband cubic and retaining $l^4$ givesFor $r=1-q^2>0$, $\mu /\sigma >(1-3q^2)/(1-q^2)$ 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.