= Charged complex sine-Gordon soliton
{title2=$\mathcal M(Q)=4M\sin(\lambda^2|Q|/4)/\lambda^2$}
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 $4[\operatorname{sgn}(\alpha)\pi/2-\alpha]/\lambda^2$, for the generator $\delta\psi=i\psi$ and $\alpha\ne0$. The conserved phase angle is a <collective coordinate> with momentum $Q$. <Quantization of a periodic soliton coordinate> gives integer $Q$ in units $\hbar=1$ 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.
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