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

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- Solution a

Solution

For the Airy equation, let

P = e^{- ik x + ω t} q

. Its local relation follows by arranging the flux so that the unwanted spatial derivatives cancel:

(\partial_{x} - ik) (- q_{xx} - ik q_{x} + k^{2} q) = - q_{xxx} - i k^{3} q .

Consequently

P_{t} - (e^{- ik x + ω t} X)_{x} = e^{- ik x + ω t} [q_{t} + q_{xxx} + (ω + i k^{3}) q] .

Thus the dispersion relation and the required flux are given by

ω (k) = - i k^{3}, X = - q_{xx} - ik q_{x} + k^{2} q .

The calculation is polynomial in the complex spectral parameter, so the local relation holds for every

k \in C

.

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