= 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 $k\Lambda_{T} = \frac{\sqrt{2 |c_{2}|c_{0}}n_{0}}{k_{B} T}$.](fig4.eps "fig:")]{} (-28, 110)

**$(\omega/c_{0}n_{0})^{2}$**

(95, 115)

**$\omega/c_{0}n_{0}$**

(-30, 35)

**$(\omega/c_{0}n_{0})^{2}$**

(92, 30)

**$(\omega/c_{0}n_{0})^{2}$**

(155, 87)[**$k \Lambda_{T}$**]{} (35, 87)[**$k \Lambda_{T}$**]{} (155, 7)[**$k \Lambda_{T}$**]{} (35, 7)[**$k \Lambda_{T}$**]{}

Linearizing about this state, and dropping the collision term, the Boltzmann equation can be written in Fourier space: $$\begin{aligned}
\label{eq:4}
(-\omega + \frac{\vec{k}\cdotp\vec{p}}{m})\tilde{\delta\textbf{F}} = \Big[\textbf{V}_{0}, \tilde{\delta\textbf{F}}\Big] + \Big[\tilde{\delta\textbf{V}}, \textbf{F}_{0}\Big] + \\\nonumber
\frac{1}{2}\Big\{\vec{k}~\tilde{\delta\textbf{V}},\vec{\nabla}_{p}\textbf{F}_{0}\Big\} \end{aligned}$$ where $\textbf{F}_{0}$ is the initial distribution and $\textbf{V}_{0}$ is the initial interaction potential. The quantities $\tilde{\delta\textbf{F}}$ and $\tilde{\delta\textbf{V}}$ are Fourier transforms of the change in the distribution and interaction, defined in the usual way $\delta f(\vec{k}, p, \omega) = \int d \vec{r}dt e^{i(\vec{k}\cdotp\vec{r} - \omega t)} \delta f(p,r, t)$. Inserting the form for the interaction potential (\[eq:3\]), 