Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 313 / 1 / ii

Let

y = x - X (t)

and substitute

ϕ = f_{E} (y) e^{i θ (t) + i u (t) y} .

Separating the coefficient of

f_{E}^{'}

and the constant and linear coefficients multiplying

f_{E}

gives

\dot{X} = u, \overset{u}{˙} = 0, \dot{θ} = - E + \frac{1}{2} u^{2} .

This is the Galilean boost of the stationary soliton.

Write

b = 2∣ E ∣

. Its conserved squared norm is

N (E) = \int_{R} f_{E}^{2} d x = 2 b .

Using

\int f_{E}^{'2} d x = \frac{2 b ^{3}}{3}, \int f_{E}^{4} d x = \frac{4 b ^{3}}{3},

gives

H = - \frac{b ^{3}}{3} + b u^{2}, P = 2 b u = N (E) u .

Therefore

H = - \frac{N ( E ) ^{3}}{24} + \frac{P ^{2}}{2 N ( E )} .

The moving soliton behaves like a classical particle of inertial mass

N (E)

, with negative internal binding energy

- N (E)^{3} /24

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