$C^8$ for which we have the following: for every $(\check w_0,z_0,a_0)\in \Xi$ there exists $\alpha,\beta\in \mathbb R$ such that if we define $w_0$ by then the conclusion of Theorem \[thm:general\] holds for initial data $(w_0,z_0,a_0)$.

Modulation variables and unstable ODEs at $x = 0$
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In order to isolate the self-similar profile, we will need to introduce modulated self-similar variables. These modulation variables allow one to control the time, location, and amplitude of the eventual shock. The idea of using modulation variables is by now classical (cf. [@Merle96; @MeZa97; @MeRa05]). We give the precise definitions of our self-similar variables and modulation variables in Section \[sec:var\], but to facilitate the forthcoming discussion, let us consider the self-similar quantities $(W, Z, A)$ defined through $w(\theta, t) = e^{- \frac{s}{4}} W(x, s) + \kappa(t),  z(\theta, t) = Z(x, s)$ and $a(\theta, t) = A(x, s)$, where we rescale time via $s =- \log( \tau - t)$ and space via $x = \frac{\theta - \xi(t)}{(\tau - t)^{\frac 5 4}}$.

In our case, we introduce the dynamical modulation variables $\tau, \xi, \kappa$ found in , to enable us to constrain $$\begin{aligned}
 \label{const:intro}
W(0, s) = 0, \qquad {\ensuremath{\partial}}_x W(0, s) = -1, \qquad {\ensuremath{\partial}}_x^4 W(0, 