W)\right) {\ensuremath{\partial}}_x Z    \label{eq:Z:0}
=  -  \beta_\tau e^{-s} A \Big(\beta_3(e^{- \frac s 4}W+ \kappa) + \beta_4Z\Big)\,,\end{aligned}$$ and as $$\begin{aligned}
 {\ensuremath{\nonumber}}&{\ensuremath{\partial}}_s A + \left(\frac 5 4 x+ \beta_{\tau}(e^{\frac 1 4 s}(\beta_1\kappa-\dot{\xi} +\beta_1Z)+\beta_1W)\right) {\ensuremath{\partial}}_x A \\ 
&\quad=  -  2\beta_\tau\beta_1e^{-s} A^2 + \frac{1}{2}\beta_\tau\beta_1  e^{-s} \Big( e^{- \frac s 4} W + \kappa + Z \Big)^2  \label{eq:A:0}
-  \beta_\tau\beta_5 e^{-s} \Big( e^{- \frac s 4} W + \kappa - Z \Big)^2\,.\end{aligned}$$

We now compactify the above equations by introducing the following transport speeds $$\begin{aligned}
 \label{gw:def}
&g_W :=\beta_{\tau}W- \beta_\tau (\dot{\xi} - \kappa) e^{\frac 1 4 s} + \beta_{\tau}\beta_2e^{\frac 1 4 s} Z
=:\beta_{\tau}W+G_W\,,\\ \label{gz:def}
&g_Z :=\beta_{\tau}\beta_2W+ \beta_{\tau}e^{\frac 1 4 s}(\beta_2\kappa-\dot{\xi} +Z)
=:\beta_{\tau}\beta_2W+G_Z \,,\\ \label{ga:def}
&g_A := \beta_{\tau}\beta_1W+ \beta_{\tau}e^{\frac 1 4 s}(\beta_1\kappa-\dot{\xi} +\beta_1Z)=:
\beta_{\tau}\beta_1W+G_A \,,  \end{aligned}$$

and forcing terms $$\begin{aligned}
 \label{def:FW}
F_W &:=-\beta_\tau e^{- \frac 3 4 s} A \Big( \beta_3 Z +\beta_4 (e^{- \frac s 4}W + \kappa) \Big)\,,\\ \label{def:FZ}
F_Z& := -  \beta_\tau e^{-s} A \Big(\beta_3(e^{- \frac s 4}W+ \kappa) + \beta_4Z\Big)\,,\\ \label{def:FA}
F_A&:=  -  2\beta_\tau\beta_1e^{-s} A^2 + \frac{1}{2}\beta_\tau\beta_1  e^{-s} \Big( e^{- \frac s 4} W + \kappa + Z \Big)^2-  \beta_\tau\beta_5 e^{-s} \Big( e^{- \frac s 4} W + \kappa - Z \Big)^2\,.\end{aligned}$$ We note that the quantities $G_W, G_Z, G_A$ are defined 