 Characteristic speed

A characteristic speed ${\lambda }_j(U)$ is an eigenvalue of the flux Jacobian $A(U)=DF(U)$ for a system of conservation laws. In a classical solution of $U_t+A(U)U_x=0$, let ${\ell }_j(U)$ be a left eigenvector. Along $\dot{x}={\lambda }_j(U(t,x))$, the chain rule givesFor a constant matrix, $w_j={\ell }_j\cdot U$ therefore obeys $(w_j{)}_t+{\lambda }_j(w_j{)}_x=0$. For example, $A=c\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ gives $w_{+}=U_1+U_2$ moving at $c$ and $w_{-}=U_1-U_2$ moving at $-c$.

For a nonlinear system, ${\ell }_j(U)\cdot dU$ need not be the differential of a scalar function, even after multiplication by a nonzero scalar factor. A scalar Riemann invariant requires such an additional integrability property. Consequently the projected identity alone does not supply a globally defined transported scalar for every wave family.