Differentiating
$$
L=-\frac d{dx}\,p^2\frac d{dx}
$$
with respect to time gives
$$
L_t=-\frac d{dx}\left(2pp_t\frac d{dx}\right).
$$
If $p_t=p^3p_{xxx}$, then
$$
L_t=-\frac d{dx}\left(2p^4p_{xxx}\frac d{dx}\right)=LA-AL.
$$
\b[Thus $L_t=[L,A]$ is an <Isospectral Lax equation>, and $(L,A)$ is a <Lax pair> for the <Harry Dym equation>.]

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 32C / c / Solution

{c}
{title2=$p_t=p^3p_{xxx}$}
{wiki}

The Harry Dym equation is the <integrable partial differential equation> $p_t=p^3p_{xxx}$. It admits a <Lax pair> based on the second-order operator $L=-\partial_xp^2\partial_x$.

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 Incoming links: Harry Dym equation