Past exam of the differential equations and applications course of the University of Oxford / 2010 / B / 5 / a / Solution

Write the system as $U_x+A(U)U_y=0$ with $U=(u,v{)}^T$ andIts two characteristic speeds are ${\lambda }_{\pm }=v\pm \sqrt{u}$, so it is strictly hyperbolic precisely when $u>0$. At $u=0$ they coincide and the matrix is not diagonalizable; for $u<0$ the speeds are nonreal.

For $u>0$ define $R_{\pm }=v\pm 2\sqrt{u}$ and $D_{\pm }={\partial }_x+(v\pm \sqrt{u}){\partial }_y$. Substitution from the two differential equations givesThereforeThese are the two Riemann invariants; their constancy applies within smooth regions, not across a shock.