Full S-matrix phase from a classical soliton delay ($d\delta /dE=\Delta T/\hslash ,S=e^{i\delta }$)

A narrow outgoing energy packet has phase $-ET/\hslash +\delta (E)$, so stationary phase gives the arrival shift $\Delta T=\hslash {\delta }^{\prime }(E)$. For two equal-mass solitons with relative rapidity $\theta$, $E=2\mathcal{M}\cosh (\theta /2)$. The convention $S=e^{2i{\delta }_{\mathrm{pw}}}$ instead gives $2\hslash {\delta }_{\mathrm{pw}}^{\prime }=\Delta T$. Classical time delays determine only the energy-dependent phase difference, not its constant branch.