 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-3/32d/d/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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)),

s

 New to topics? Read the docs here!