 Solution

In smooth regions the characteristic equations are $\dot{x}=q^{\prime }(\rho )={\rho }^3$ and $\dot{\rho }=0$. If $\xi$ labels the initial point, their parametrization isFor the two constant states, characteristics from the left have speed one and those from the right speed zero; they would overlap. The entropy solution replaces the overlap by a compressive shock. Its speed isThusThe left parametrization $x=\xi +t$ is retained only for $x<t/4$, and the right parametrization $x=\xi$ only for $x>t/4$. The characteristic speeds obey $0<c<1$, so characteristics enter the shock from both sides.