always stable. The dynamics of fluctuations in $\delta \rho $, $\tau _{y}$ and $\delta Q_{yy}$ is governed by three coupled equations. Fluctuations in $\tau _{y}$ are always quickly damped near the mean-field transition, while the decay rate of $\delta Q_{yy}$, controlled to leading order by $\alpha _{0}$, vanishes as $\rho _{0}\rightarrow \rho _{c}^{+}$. We therefore neglect fluctuations in $\tau _{y}$ and simply examine the coupled dynamics of $\delta \rho $ and $\delta Q\equiv \delta Q_{yy}$, given by

$$\begin{gathered}
\partial _{t}\delta \rho _{\mathbf{k}}=-k^{2}\delta \rho _{\mathbf{k}}-
\overline{D}_Qk^{2}\delta Q_{\mathbf{k}}  \label{6.1} \\
\partial _{t}\delta Q_{\mathbf{k}}=-\left[\frac{ \alpha _{0}}{2}+(1+\overline{D}
_{s})k^{2}\right] \delta Q_{\mathbf{k}}-\frac{1}{2}(\alpha ^{\prime
}Q_{0}+\overline{D}_\rho k^{2})\delta \rho _{\mathbf{k}}  \label{6.2}\end{gathered}$$

where $\alpha ^{\prime }=\left( \frac{\partial \alpha }{\partial \rho }%
\right) _{\rho =\rho _{0}}$, or $\alpha ^{\prime }=1/\rho _{c}$ with the chosen parameters. The decay of density and ordered parameter fluctuations is then controlled by two coupled hydrodynamic modes. One of the modes has a finite decay rate (proportional to $\alpha _{0}$) in the limit $k\rightarrow
0$ and is always stable. At small wavevector, the dispersion relation of the other mode is given by

$$s_{y}(k)=-s_{2}k^{2}-s_{4}k^{4}+\mathcal{O}(k^{6})\;.  \label{7}$$

with $s_{2}=1-\frac{\overline{D}\alpha ^{\prime }}{\sqrt{\alpha _{0}\beta }}$ and $s_{4}>0$. Near the transition where $\alpha _{0}\rightarrow 0$, $s_{2}<0
$ and $s_{4}\simeq \frac{2\overline{D}^{2}\alpha ^{^{\prime }2}}{\alpha
_{0}^{2}\beta 