Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 67 / 1 / a / Solution

For the Airy equation, let $P=e^{-ikx+\omega t}q$. Its local relation follows by arranging the flux so that the unwanted spatial derivatives cancel:ConsequentlyThus the dispersion relation and the required flux are given byThe calculation is polynomial in the complex spectral parameter, so the local relation holds for every $k\in \mathbb{C}$.