unstable mode remains largely unchanged. The experimental consequence is that $q<0$, one should observe a reduction in the size of ferromagnetic domains, while leaving the timescale for the onset of the instability unaffected.

(150, 90)(10, 10) (-165, 100)[(a)]{} (-35, 100)[(b)]{} (95, 100)[(c)]{} (220, 100)[(d)]{} (-160, 15)[![image](fig17.eps)]{} (-35, 15)[![image](fig16.eps)]{} (95, 15)[![image](fig22.eps)]{} (220, 15)[![image](fig23.eps)]{} (-45, 42)

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

(84, 42)

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

(213, 42)

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

(-170, 42)

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

(200, 7)[**$k \Lambda_{T}$**]{} (70, 7)[**$k \Lambda_{T}$**]{} (-60, 7)[**$k \Lambda_{T}$**]{} (325, 7)[**$k \Lambda_{T}$**]{}

*Large anti-ferromagnetic interactions*: The modes of the polar gas, when $c_{2} > 0$ can be obtained from (\[eq:8\]). For small $c_{2}/c_{0}$, the polar state is stable toward spin and nematic fluctuations. However, when $c_{2}$ and $c_{0}$ are commensurate, an instability can set in. This latter limit, which has been largely unexplored, is primarily of academic interest as all current spinor gas experiments have $|c_{2}| \ll c_{0}$. For simplicity we restrict our analysis to the case of no magnetic field ($q=0$).

As shown in Fig. \[fig:-3\](a), for $c_{2} = 0$ we find a gapped and ungapped quadratic mode with dispersions $\omega(k) = \sqrt{\frac{1}{2}({\Omega^{2}_{d} \pm \sqrt{ \Omega^{4}_{d} - (v_{T}k)^{4}}})}$, where $\Omega_{d}$ reduces to $c_{0}n_{0}$ in this limit. As $c_{2}$ is increased, for some $k_{c}$, the gapped 