of the shock development, we will rewrite - in terms of the Riemann invariants $$w= b+  {\frac{1}{\lambda }} P^\lambda \,, \qquad z= b- {\frac{1}{\lambda }}  P^\lambda \,,$$ and the wave speeds $$\Lambda_1= b - P^\lambda= \frac{1- \lambda }{2} w +  \frac{1+ \lambda }{2} z \,, \qquad  \Lambda_2= b+ P^ \lambda =  \frac{1+ \lambda }{2} w +  \frac{1- \lambda }{2} z\,.$$ With these substitutions we obtain the following system of nonlinear transport equations

\[eq:euler:wza\] $$\begin{aligned}
\partial_t w +\left(w+\tfrac{1-\lambda}{1+\lambda}z\right)\partial_{\theta}w 
&= -a  \left(\tfrac{1-2\lambda}{1+\lambda} z+ \tfrac{3+2\lambda}{1+\lambda} w\right) 
\label{eq:w:evo} \,,\\
\partial_t z +\left(z+\tfrac{1-\lambda}{1+\lambda}w\right)\partial_{\theta}z 
&=  -a  \left( \tfrac{1-2\lambda}{1+\lambda} w+ \tfrac{3+2\lambda}{1+\lambda} z\right)
\label{eq:z:evo} \,, \\
\partial_t a +\tfrac{1}{1+\lambda} (w+z) \partial_{\theta}a  
&=-\tfrac{2}{1+\lambda}a^2+\tfrac{1}{2(1+\lambda)}(w+z)^2 - \tfrac{\lambda}{2(1+\lambda)}(w-z)^2   \,.
\label{eq:a:evo}\end{aligned}$$

Initial data assumptions {#ss:initial}
------------------------

In this section we will describe the initial data used to construct unstable shock solutions. We introduce a large constant $M$ which will be used to bound certain implicit constants appearing in the paper. We also let $\eps>0$ be a small constant which will parameterize the slope of the initial data.

We will denote the initial data at initial time $t=-\eps$ by $$w(\theta,-\eps)=w_0,\quad z(\theta,-\eps)=z_0, \quad a(\theta,-\eps)=a_0\,.$$ The initial will be assumed to satisfy the follow support assumptions $${\ensuremath{\mathrm{supp\,}}}(w_0-\kappa_0)\cup {\ensuremath{\mathrm{supp\,}}}(z_0)\cup{\ensuremath{\mathrm{supp\,}}}(a_0) \subset\left[ -\frac{M\eps}{2} ,\frac{M\eps}{2}\right]\,,$$ where $\kappa_0>0$ will be a 