Why is the partial differential equation $u_{t} + a u_{x} = 0$ hyperbolic?

Question

I don't quite understand why this advection equation, with some initial condition $u(0,x) = u_{0}(x)$, is considered hyperbolic (as for instance here). If I apply the test mentioned in Farlow, comparing it to $$Au_{xx}+Bu_{xy} + Cu_{yy} + D u_{x} + Eu_{y} + Fu = G$$ then both $B^{2}$ and $AC$ are zero, and thus $B^{2} - 4AC = 0 \Rightarrow \text{parabolic}$. Or does a different criterion apply when there aren't any second-order terms?

Answer 1

You have to look at the principal symbol $$ a_1(x,t,\xi,\tau)=i\tau+ai\xi. $$ For every fixed $\xi$ the equation $a_1(x,t,\xi,\tau)=0$ has exactly one real solution in $\tau$. Therefore the equation is hyperbolic with respect to $t$. Basically, this means that there is a real characteristic passing through each point. You can read more on the topic here: https://people.sissa.it/~dabrow/IFM12small1.pdf