 Charged complex sine-Gordon soliton

The rest-frame field $\psi =\cos \alpha e^{i\sin \alpha t}/[\lambda \cosh (\cos \alpha x)]$ has mass $4M\cos \alpha /{\lambda }^2$ and charge $4[\mathrm{sgn}(\alpha )\pi /2-\alpha ]/{\lambda }^2$, for the generator $\delta \psi =i\psi$ and $\alpha \ne 0$. The conserved phase angle is a collective coordinate with momentum $Q$. Quantization of a periodic soliton coordinate gives integer $Q$ in units $\hslash =1$ and the displayed leading mass formula. On the regular branch $0<|Q|<2\pi /{\lambda }^2$, its concave increasing sine law prevents fragmentation into smaller like-charge states.