Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 105 / 1 / i / Solution

Characteristic. The third-order principal symbol of ${\partial }_t-{\partial }_x^3$ is $-{\nu }_x^3$. The initial line has normal $({\nu }_x,{\nu }_t)=(0,1)$, on which this symbol vanishes. Its evolution form does not make it non-characteristic in the ordinary total-order definition.