 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-67/1/a/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

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 .

k \in C

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