This in turn cannot be enforced by the introduction of further modulation variables due to the lack of further symmetries in the compressible Euler equations, and so must be enforced by the choice of the unstable manifold, $\mathcal{M}_U$.

The equations describing the second and third derivatives of $W$ at $x = 0$ are given by $$\begin{aligned}
 \label{super:1}
&({\ensuremath{\partial}}_s - \frac 3 4) W^{(2)}(0, s) = \text{rapidly decaying forcing terms}\,, \\ \label{super:2}
&({\ensuremath{\partial}}_s - \frac{1}{2}) W^{(2)}(0, s) = \text{rapidly decaying forcing terms}\,.\end{aligned}$$ One can see the instability of the manifold due to the negative damping coefficients appearing on the left-hand side of - . Indeed, negatively damped ODEs such as - generically grow as $s \rightarrow \infty$, but certain data (as determined by the right-hand side) can lead to decaying solutions.

In the context of the Euler equations, the right-hand sides above themselves depend on other elements of the system (such as the modulation variables, and other derivatives of $(W, Z, A)$). For this reason, we are led to develop a Newton scheme which identifies this unstable manifold.

An iterative scheme to search for unstable solutions
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In this subsection, we briefly discuss the Newton scheme that can be used to 