Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 34C (Integrable Systems)

The stated operators form a Lax pair when the KdV equation is equivalent toIndeed, if $L\varphi =k^2\varphi$, differentiation and this identity showThus ${\partial }_t+A$ preserves every spectral eigenspace, which is the Isospectral Lax equation formulation.

Translation by $2\pi$ preserves the scattering equation, so it acts linearly on its two-dimensional solution space. Because $u$ is real and the initial conditions are conjugate,
${\phi }_{-}=\overline{{\phi }_{+}}$, forcing the transfer matrix to have the formThe Wronskian of two solutions of
$-{\varphi }^{\prime \prime }+u\varphi =k^2\varphi$ is independent of $x$. Translation therefore preserves it, soConsequently

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.