Hyperbolic conservation law

Remarks

It is strictly hyperbolic if moreover the eigenvalues are distinct, $\lambda_1 < \cdots < \lambda_m$. In one space dimension $B(u; \xi) = \xi\, DF(u)$, and the condition is that $DF(u)$ is diagonalizable with real eigenvalues; these are the characteristic speeds. The definition makes the system a hyperbolic system of first-order equations when written in the non-conservative form $u_t + \sum_j DF_j(u)\, u_{x_j} = 0$.

Example: every scalar conservation law ($m = 1$) is hyperbolic, since $B(u; \xi) = F'(u) \cdot \xi$ is a real $1 \times 1$ matrix. Example: the $p$-system $v_t - w_x = 0$, $w_t + p(v)_x = 0$ with $p' < 0$ has $DF = \begin{pmatrix} 0 & -1 \\ p'(v) & 0 \end{pmatrix}$. Its eigenvalues $\pm \sqrt{-p'(v)}$ are real and distinct, so the system is strictly hyperbolic.

Non-example: the linear system $u_t + A u_x = 0$ with $A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$, so that $F(u) = Au$. The eigenvalues of $A$ are $\pm i$, which are not real, so the system is not hyperbolic; its components satisfy the Cauchy–Riemann equations, which are elliptic.