 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-83/1/iv/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

For a straight unit-winding quantum vortex, the real radial amplitude satisfies

R^{''} + \frac{1}{s} R^{'} - \frac{R}{s ^{2}} + (1 - R^{2}) R = 0, R (0) = 0, R (\infty) = 1.

In the frame translating with the ring, put

Z = z - u t

and

Φ = e^{i θ} (R + ϕ)

, where

ϕ

is independent of the angular coordinate. The cylindrical Laplacian has the exact identity

\nabla^{2} [e^{i θ} (R + ϕ)] = e^{i θ} [\frac{1}{s} \partial_{s} (s \partial_{s} (R + ϕ)) + \partial_{Z}^{2} ϕ - \frac{R + ϕ}{s ^{2}}] .

After inserting this into the translating Gross–Pitaevskii equation, subtract the straight-vortex equation. Because

R

is real,

∣ R + ϕ ∣^{2} = R^{2} + R (ϕ + ϕ^{*}) + ∣ ϕ ∣^{2},

and direct expansion of the nonlinear difference gives

(1 - ∣ R + ϕ ∣^{2}) (R + ϕ) - (1 - R^{2}) R = (1 - 2 R^{2}) ϕ - R^{2} ϕ^{*} - R ϕ^{2} - 2 R ∣ ϕ ∣^{2} - ∣ ϕ ∣^{2} ϕ .

Thus the exact ring-on-line perturbation equation, including its quadratic and cubic terms, is

2 i u ϕ_{Z} = \frac{1}{s} \partial_{s} (s ϕ_{s}) + ϕ_{ZZ} - \frac{ϕ}{s ^{2}} + [1 - 2 R^{2} - R (ϕ + 2 ϕ^{*}) - ∣ ϕ ∣^{2}] ϕ - R^{2} ϕ^{*} .

The two appearances of

R ∣ ϕ ∣^{2}

are responsible for the coefficient

2

multiplying

ϕ^{*}

inside the bracket. No small-amplitude approximation is needed for this identity.

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