Let $y=x-X(t)$ and substitute
$$
\phi=f_E(y)e^{i\theta(t)+iu(t)y}.
$$
Separating the coefficient of $f_E'$ and the constant and linear coefficients multiplying $f_E$ gives
$$
\boxed{\dot X=u,
\qquad\dot u=0,
\qquad\dot\theta=-E+\frac12u^2}.
$$
This is the <Galilean boost> of the stationary soliton.

Write $b=\sqrt{2|E|}$. Its conserved squared norm is
$$
N(E)=\int_{\mathbb R}f_E^2dx=2b.
$$
Using
$$
\int f_E'^2dx=\frac{2b^3}{3},
\qquad
\int f_E^4dx=\frac{4b^3}{3},
$$
gives
$$
\boxed{H=-\frac{b^3}{3}+bu^2},
\qquad
\boxed{P=2bu=N(E)u}.
$$
Therefore
$$
\boxed{H=-\frac{N(E)^3}{24}+\frac{P^2}{2N(E)}}.
$$
The moving soliton behaves like a classical particle of inertial mass $N(E)$, with negative internal binding energy $-N(E)^3/24$.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-313/1/ii/solution