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

Treat $\varphi$ and $\bar{\varphi }$ as independent fields. The Lagrangian density isThe Euler-Lagrange equation for $\bar{\varphi }$ isorThe equation from variation of $\varphi$ is its complex conjugate.

Put $\mu =|E|=-E>0$ and seek $\varphi =e^{-iEt}f(x)$ with real $f$. The equation becomesFor $b=\sqrt{2\mu }$, the identityshows thatsatisfies the stationary equation. Hence $e^{-iEt}{f}_E(x)$ is the bright soliton of the focusing nonlinear Schrödinger equation.

Let $y=x-X(t)$ and substituteSeparating the coefficient of $f_E^{\prime }$ and the constant and linear coefficients multiplying $f_E$ givesThis is the Galilean boost of the stationary soliton.

Write $b=\sqrt{2|E|}$. Its conserved squared norm isUsinggivesThereforeThe moving soliton behaves like a classical particle of inertial mass $N(E)$, with negative internal binding energy $-N(E{)}^3/24$.

Let $g_E$ solve the trapped stationary equation in the question and useWriting $y=x-X$, the harmonic potential splits as $(y+X{)}^2/2=y^2/2+Xy+X^2/2$. Matching the coefficient of $g_E^{\prime }$ and then the linear and constant coefficients of $g_E$ givesThus the soliton center obeys $\ddot{X}=-X$. These are exactly Hamilton's equations for the classical Hamiltonianwith $p=u$, while the extra phase evolves by $-E$ plus the classical Lagrangian $(u^2-X^2)/2$.

Use the four-parameter ansatz from part (ii), now allowing every parameter to vary slowly. Oddness of $y{f}_E(y{)}^2$ and the stationary-profile integrals give the collective-coordinate effective LagrangianwhereVariation of $\theta$ makes $N(E)$, and hence $E$, constant. Variation of $u$, $X$, and $E$ then givesThus the center follows the classical Hamiltonianto leading adiabatic order. The first neglected profile correction is of order ${ϵ}^3$, because the term linear in $y$ integrates to zero.