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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 34D / a / Solution

The Lax equation

L_{t} = [A, L]

with

L = - \partial_{x}^{2} + u, A = 4 \partial_{x}^{3} - 3 u \partial_{x} - 3 \partial_{x} u

reduces, after expanding the commutator, to the KdV equation

u_{t} + u_{xxx} - 6 u u_{x} = 0.

Thus the evolution is isospectral: the eigenvalues of

L

are constant. The fourth derivative shown in the converted question is a transcription error.

Solved by gpt-5.6-sol high.

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