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Sound waves and modulational instabilities on continuous wave solutions in spinor Bose-Einstein condensates

We analyze sound waves (phonons, Bogoliubov excitations) propagating on continuous wave (cw) solutions of repulsive $F=1$ spinor Bose-Einstein condensates (BECs), such as $^{23}$Na (which is antiferromagnetic or polar) and $^{87}$Rb (which is ferromagnetic). Zeeman splitting by a uniform magnetic field is included. All cw solutions to ferromagnetic BECs with vanishing $M_F=0$ particle density and non-zero components in both $M_F=\pm 1$ fields are subject to modulational instability (MI). MI increases with increasing particle density. MI also increases with differences in the components' wavenumbers; this effect is larger at lower densities but becomes insignificant at higher particle densities. CW solutions to antiferromagnetic (polar) BECS with vanishing $M_F=0$ particle density and non-zero components in both $M_F=\pm 1$ fields do not suffer MI if the wavenumbers of the components are the same. If there is a wavenumber difference, MI initially increases with increasing particle density, then peaks before dropping to zero beyond a given particle density. The cw solutions with particles in both $M_F=\pm 1$ components and nonvanishing $M_F=0$ components do not have MI if the wavenumbers of the components are the same, but do exhibit MI when the wavenumbers are different. Direct numerical simulations of a cw with weak white noise confirm that weak noise grows fastest at wavenumbers with the largest MI, and shows some of the results beyond small amplitude perturbations. Phonon dispersion curves are computed numerically; we find analytic solutions for the phonon dispersion in a variety of limiting cases.