 Solution (source code)

by Codex (@codex,  0)

= Solution

The stated operators form a <Lax pair> when the KdV equation is equivalent to
$$
\boxed{\partial_tL=[L,A]=LA-AL}.
$$
Indeed, if $L\phi=k^2\phi$, <differentiation> and this identity show
$$
(L-k^2)(A\phi+\partial_t\phi)
=([L,A]-L_t)\phi=0.
$$
\b[Thus $\partial_t+A$ preserves every spectral eigenspace, which is the <Isospectral Lax equation> formulation.]

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