through the second equalities in - .

With these definitions, our equations become $$\begin{aligned}
 \label{W:0}
({\ensuremath{\partial}}_ s- \frac 1 4) W + (g_W + \frac{5}{4}x ) {\ensuremath{\partial}}_x W &= -e^{- \frac 3 4 s} \frac{\dot{\kappa}}{1 - \dot{\tau}} +  F_W\,, \\ \label{Z:0}
{\ensuremath{\partial}}_s Z + (g_Z + \frac 5 4 x ) {\ensuremath{\partial}}_x Z &= F_Z\,, \\ \label{A:0}
{\ensuremath{\partial}}_s A + (g_A + \frac 5 4 x) {\ensuremath{\partial}}_x A &= F_A\,. \end{aligned}$$

Further, it will be convenient to introduce the notation $$\mathcal V_W:=g_W + \frac{5}{4}x,\qquad
\mathcal V_Z:=g_Z + \frac{5}{4}x,\qquad
\mathcal V_A:=g_A + \frac{5}{4}x\,.$$ so that we obtain $$\begin{aligned}
 \label{basic:w}
({\ensuremath{\partial}}_ s- \frac 1 4) W + \mathcal V_W {\ensuremath{\partial}}_x W &= -e^{- \frac 3 4 s} \frac{\dot{\kappa}}{1 - \dot{\tau}} +  F_W\,, \\ \label{basic:z}
{\ensuremath{\partial}}_s Z + \mathcal V_Z {\ensuremath{\partial}}_x Z &= F_Z\,, \\ \label{basic:a}
{\ensuremath{\partial}}_s A +  \mathcal V_A {\ensuremath{\partial}}_x A &= F_A\,. \end{aligned}$$

We define now the combination $$\begin{aligned}
 \label{def:mu}
\mu :=- \beta_{\tau} (\dot{\xi} - \kappa)e^{\frac s 4} +\beta_{\tau}\beta_2e^{\frac14 s}Z(0,s) = G_W(s, 0)\,. \end{aligned}$$

An unstable self-similar solution to Burgers’ equation {#s:burgers}
------------------------------------------------------

Here we develop properties of the self-similar Burgers profile, ${\overline}{W}:= {\overline}{W}_2$, which solves the equation $$\begin{aligned}
 \label{Burger:1}
- \frac 1 4 {\overline}{W} + ({\overline}{W} + \frac 5 