s) = 0\,,  \end{aligned}$$ where the final constraint is notably different than in the works [@BuShVi2019; @BuShVi2019b; @BuShVi2019c], and reflects the different nature of the self-similar profile ${\overline}{W}_2$.

In so doing, we obtain from - the system that we ultimately analyze, which $$\begin{aligned}
 \label{W:0:i}
({\ensuremath{\partial}}_ s- \frac 1 4) W + (g_W + \frac{5}{4}x ) {\ensuremath{\partial}}_x W &= -e^{- \frac 3 4 s} \frac{\dot{\kappa}}{1 - \dot{\tau}} +  F_W, \\ \label{Z:0:i}
{\ensuremath{\partial}}_s Z + (g_Z + \frac 5 4 x ) {\ensuremath{\partial}}_x Z &= F_Z\,, \\ \label{A:0:i}
{\ensuremath{\partial}}_s A + (g_A + \frac 5 4 x) {\ensuremath{\partial}}_x A &= F_A\,. \end{aligned}$$ Above, the quantities $g_W, g_Z, g_A$ are transport speeds, and $F_W, F_Z, F_A$ are forcing terms that we also leave unspecified for the purposes of this discussion. The reader may find the precise definitions in - and - .

In addition, we control the evolution of $\tau, \xi, \kappa$ through ODEs obtained by restricting to the constrains, . Importantly the three modulation variables enable us to constrain only the three quantities appearing in . However, a feature of with $i \ge 2$ is that $W^{(2)}(0, s)$ and $W^{(3)}(0, s)$ need to be zero in the limit as $s \rightarrow \infty$. 