1.5.3 Hyperbolic system

A linear system (43) of PDEs is called hyperbolic if $\bgroup\color{DEFcolor}$\bar{\bar{\ensuremath{\mathsf{A}}}}$\egroup$ is diagonalizable, i.e., there exists a matrix $\bgroup\color{DEFcolor}$\bar{\bar{\ensuremath{\mathsf{Q}}}}$\egroup$ with

$\begin{displaymath} \bar{\bar{\ensuremath{\mathsf{\Lambda}}}} = \bar{\bar{\ensu... ...{\ensuremath{\mathsf{A}}}} \bar{\bar{\ensuremath{\mathsf{Q}}}} \end{displaymath}$

(46)

and $\bgroup\color{DEFcolor}$\bar{\bar{\ensuremath{\mathsf{\Lambda}}}}$\egroup$ is in diagonal form (with real numbers on the diagonal: the eigenvalues of $\bgroup\color{DEFcolor}$\bar{\bar{\ensuremath{\mathsf{A}}}}$\egroup$ ).

With the definition

$\begin{displaymath} \ensuremath{\mathchoice{\mbox{\boldmath$\displaystyle q$}} ... ...h$\scriptstyle q$}} {\mbox{\boldmath$\scriptscriptstyle q$}}} \end{displaymath}$

(47)

Eq. (43) gets the characteristic form

$\begin{displaymath} \;\;\!\! \frac{\partial }{\partial t} \ensuremath{\mathchoi... ...yle 0$}} {\mbox{\boldmath$\scriptscriptstyle 0$}}} \enspace . \end{displaymath}$

(48)

This is now a set of independent equations, each of the simple form

$\begin{displaymath} \;\;\!\! \frac{\partial }{\partial t} q'_i + \lambda_i \; \frac{\partial }{\partial x} q'_i = 0 \enspace . \end{displaymath}$

(49)

A quasi-linear system with $\bgroup\color{DEFcolor}$\bar{\bar{\ensuremath{\mathsf{A}}}} \left( \ensuremath{\... ...scriptstyle q$}} {\mbox{\boldmath$\scriptscriptstyle q$}}}, x, t \right)$\egroup$ can be hyperbolic at point $\bgroup\color{DEFcolor}$\left( \ensuremath{\mathchoice{\mbox{\boldmath$\displays... ...scriptstyle q$}} {\mbox{\boldmath$\scriptscriptstyle q$}}}, x, t \right)$\egroup$ .