3.1.7 Similarity solutions I

The quasi-linear PDE

$\begin{displaymath} \frac{\partial q}{\partial t} + \frac{\mathrm{d} f}{\mathrm{d} q} \frac{\partial q}{\partial x} = 0 \end{displaymath}$

(173)

for the Riemann problem

$\begin{displaymath} q \! \left( x, 0 \right) = \left\{ \begin{array}{ll} q_\... ... q_\mathrm{r} & \mbox{if} \enspace x > 0 \end{array} \right. \end{displaymath}$

(174)

has similarity solutions of the form

$\begin{displaymath} q \! \left( x, t \right) = \tilde{q} \! \left( \frac{x}{t} \right) \enspace . \end{displaymath}$

(175)