 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-2/34c/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

The stated operators form a Lax pair when the KdV equation is equivalent to

\partial_{t} L = [L, A] = L A - A L .

Indeed, if

L ϕ = k^{2} ϕ

, differentiation and this identity show

(L - k^{2}) (A ϕ + \partial_{t} ϕ) = ([L, A] - L_{t}) ϕ = 0.

Thus $\partial_{t} + A$ preserves every spectral eigenspace, which is the Isospectral Lax equation formulation.

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