 Periodic KdV transfer matrix (source code)

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= Periodic KdV transfer matrix

For real periodic KdV data, translation by one period acts on a conjugate basis of scattering solutions by
$$
T=\begin{pmatrix}a&b\\\bar b&\bar a\end{pmatrix},
\qquad |a|^2-|b|^2=1.
$$
Lax compatibility gives $\dot T=[\Lambda,T]$ for the matrix of $\partial_t+A$ on the eigenspace. Hence $\operatorname{tr}T=2\operatorname{Re}a$ is conserved.

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