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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-84/2/i/solution

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Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 84 / 2 / i / Solution by

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

To first order,

∣1 + u + i v ∣^{2} = 1 + 2 u

and

(1 - ∣ ψ ∣^{2}) ψ = - 2 u

. Equating real and imaginary parts of the Gross–Pitaevskii equation therefore gives

2 v_{t} = \nabla^{2} u - 2 u, - 2 u_{t} = \nabla^{2} v .

Differentiate the second equation in time and use the first to eliminate

v

:

u_{tt} = \frac{1}{2} \nabla^{2} u - \frac{1}{4} \nabla^{4} u .

For a Fourier mode with

k = ∣ k ∣

, the dispersion relation is

- ω^{2} = - \frac{k ^{2}}{2} - \frac{k ^{4}}{4}, ω_{\pm} (k) = \pm \frac{k}{2} k^{2} + 2 .

This is the normalized Bogoliubov spectrum. At small wavenumber the positive branch has sound speed

lim_{k \to 0} ω_{+} (k) / k = 1/ 2

; at large wavenumber it approaches the free-particle quadratic dispersion. The linear frequencies are real, confirming stability of the repulsive uniform bulk.

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