Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 32B (Integrable Systems)

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- Solution 32B

Solution

The flow of

W = ϕ u \partial_{u} + \sum α_{i} x_{i} \partial_{x_{i}}

is

(u, x_{i}) \mapsto (e^{ϕ s} u, e^{α_{i} s} x_{i})

; invariants are exactly solutions of

W f = 0

. Two scaling symmetries of the PDE are

W_{1} = 2 u \partial_{u} + x \partial_{x} - t \partial_{t}, W_{2} = x \partial_{x} + y \partial_{y} + t \partial_{t} .

Invariants for

W_{1}

include

y, x t, u t^{2}

; for

W_{2}

,

u, y / x, t / x

. Two common invariants are

u t / x

and

y^{2} / (x t)

. Invariance under

u \partial_{u} + x \partial_{x} + \frac{1}{2} y \partial_{y}

and

\partial_{y}

u = x F (t)

. Substitution gives

F^{'} = F^{2}

, hence

u = 0

or

u = - x / (t - C)

.

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