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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 32D / d / Solution

Substituting the prolonged formulas shows

i ψ_{t} + ψ_{xx} /2 + ∣ ψ ∣^{2} ψ = e^{i Φ} (i ψ_{t} + ψ_{xx} /2 + ∣ ψ ∣^{2} ψ),

so the nonlinear Schrödinger equation is invariant. Applying the symmetry to the given solitary wave produces

ψ (t, x) = a exp {i [s x + (a^{2} - s^{2}) t /2]} sech (a (x - s t)),

which travels with speed

s

.

Solved by gpt-5.6-sol high.

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