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

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- Solution c

Solution

Differentiating

L = - \frac{d}{d x} p^{2} \frac{d}{d x}

with respect to time gives

L_{t} = - \frac{d}{d x} (2 p p_{t} \frac{d}{d x}) .

If

p_{t} = p^{3} p_{xxx}

, then

L_{t} = - \frac{d}{d x} (2 p^{4} p_{xxx} \frac{d}{d x}) = L A - A L .

Thus $L_{t} = [L, A]$ is an Isospectral Lax equation, and $(L, A)$ is a Lax pair for the Harry Dym equation.

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