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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 8A / iv / Solution

The dark soliton density is

∣ ψ_{0} (ξ) ∣^{2} = 1 - (1 - 2 U^{2}) sech^{2} (\frac{1 - 2 U ^{2}}{2} ξ) .

For

U = 0

,

∣ ψ_{0} (ξ) ∣^{2} = tanh^{2} (ξ / 2),

a stationary notch reaching zero at its centre. For

U = 1/2

,

∣ ψ_{0} (ξ) ∣^{2} = 1 - \frac{1}{2} sech^{2} (ξ /2),

a shallower notch with minimum

1/2

. Since

ξ = x - U t

, the first profile remains fixed and the second translates rigidly to the right at speed

1/2

.

Solved by gpt-5.6-sol high.

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