= Conserved-field sideband dispersion relation
Linearize around $A=Re^{iqX}$ 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 under
$$
M_l=\begin{pmatrix}-l^2-2r&-2ql&-2\\-2ql&-l^2&0\\-\mu r l^2&0&-\sigma l^2\end{pmatrix}.
$$
Expanding $\det(\lambda I-M_l)$ gives
$$
(\lambda+\sigma l^2)[(\lambda+l^2)(\lambda+l^2+2r)-4q^2l^2]-2\mu r l^2(\lambda+l^2)=0.
$$
At $l=0$, 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.
Back to article page
