|{\overline}{W}^{(n)}| \le C_k \eta_{- \frac 1 5 - \frac{n}{4}} \,,\\ \label{truth:1}
-1 + \frac{l^7}{50} \le {\overline}{W}^{(1)} \le 0 \quad\text{ for } |x| \ge \ell\,.\end{gathered}$$

Higher order $x$ derivatives
----------------------------

In this section we list the higher order derivatives of $(W,Z,A)$. It will be convenient to introduce the notation: $$\begin{aligned}
f^{(n)}(s, x) := {\ensuremath{\partial}}_x^n f(s, x)\,. \end{aligned}$$

We will derive now up to eight derivatives of the above system. $$\begin{aligned}
 \label{W:n}
\Big( {\ensuremath{\partial}}_s + \frac 1 4 (- 1 + 5n) + \beta_{\tau}(n+1_{n> 1}) W^{(1)}  \Big) W^{(n)} +\mathcal V_W {\ensuremath{\partial}}_x W^{(n)} &= F_{W,n}\,, \\ \label{Z:n}
({\ensuremath{\partial}}_s + \frac{5n}{4} +  n\beta_{\tau}\beta_2 W^{(1)}) Z^{(n)} + \mathcal V_Z {\ensuremath{\partial}}_x Z^{(n)} &= F_{Z, n}\,, \\ \label{A:n}
({\ensuremath{\partial}}_s + \frac 5 4 n + n\beta_{\tau}\beta_1 W^{(1)} ) A^{(n)} +\mathcal V_A{\ensuremath{\partial}}_x A^{(n)} &= F_{A,n}\,,\end{aligned}$$ where the forcings are defined by $$\begin{aligned}
 \label{F.W.n}
&F_{W,n} := F_W^{(n)} - 1_{n \ge 3}\beta_\tau \sum_{j = 2}^{n-1} \binom{n}{j} W^{(j)} W^{(n+1 - j)} - \sum_{j = 1}^n \binom{n}{j} G_W^{(j)} W^{(n+1-j)}\,, \\ \label{F.Z.n}
&F_{Z, n} := F_Z^{(n)} - 1_{n \ge 2}\beta_\tau \beta_2 \sum_{j = 2}^{n} \binom{n}{j}  W^{(j)} Z^{(n+1 - j)} - \sum_{j = 1}^n \binom{n}{j} G_Z^{(j)} Z^{(n+1-j)}\,, \\ \label{F.A.n}
&F_{A,n} := F_A^{(n)} -1_{n \ge 2} \beta_\tau \beta_1\sum_{j = 2}^{n} \binom{n}{j}  W^{(j)} A^{(n+1 - j)}- 