Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 34C / b / ii / Solution

By part (a), each $A{\phi }_{\pm }+{\partial }_t{\phi }_{\pm }$ lies in the same two-dimensional eigenspace. Reality and conjugacy again force the coefficient matrix to have the formBecause $u$ and hence $A$ are periodic, ${\partial }_t+A$ commutes with translation by $2\pi$. Applying it to
$\Psi (x+2\pi )=\hat{T}\Psi (x)$ givesThe basis $\Psi$ is independent, so the periodic KdV transfer matrix obeysTaking traces,Hence $\boxed{\mathrm{Re}a}$ is independent of time.

Solved by gpt-5.6-sol high.