line is $\omega^{2} = q(q - 4|c_{2}|n_{0})$, which is the gap predicted by the small $k$ expansion (\[eq:7\]). (d) $q=0$ dispersion in detail. Horizontal dashed line at $\omega^{2} = -4c^{2}_{2}n^{2}_{0}$ indicates the most unstable mode. Vertical dotted line is at $k\Lambda_{T} = \frac{\sqrt{2 |c_{2}|c_{0}}n_{0}}{k_{B} T}$.](fig2.eps "fig:")]{} (-25, 15)[![\[fig:-1\](Color Online) (a) Dispersion relation $\omega^2(k)$ for various values of $q$ using $^{87}$Rb parameters ($c_{2} < 0$) at $T= 1\mu$K and $n_{0} = 10^{14}$cm$^{-3}$: q=0 (solid, black), $q = 2|c_{2}|n_{0}$ (blue, dashed), $q = 4|c_{2}|n_{0}$ (green, dotted), $q = 4.5|c_{2}|n_{0}$ (red, thin). (b) Gapless, linear dispersion for small $k$ at $q = 4|c_{2}|n_{0}$. The solid black curve is the small $k$ expansion (\[eq:7\]). (c) $q = 2|c_{2}|n_{0}$ dispersion in detail. The horizontal black line is $\omega^{2} = q(q - 4|c_{2}|n_{0})$, which is the gap predicted by the small $k$ expansion (\[eq:7\]). (d) $q=0$ dispersion in detail. Horizontal dashed line at $\omega^{2} = -4c^{2}_{2}n^{2}_{0}$ indicates the most unstable mode. Vertical dotted line is at $k\Lambda_{T} = \frac{\sqrt{2 |c_{2}|c_{0}}n_{0}}{k_{B} T}$.](fig3.eps "fig:")]{} (95, 15)[![\[fig:-1\](Color Online) (a) Dispersion relation $\omega^2(k)$ for various values of $q$ using $^{87}$Rb parameters ($c_{2} < 0$) at $T= 1\mu$K and $n_{0} = 10^{14}$cm$^{-3}$: q=0 (solid, black), $q = 2|c_{2}|n_{0}$ (blue, dashed), $q 