= Sine-Gordon breather fusion amplitude
{c}
{title2=$S_{2,1}(\vartheta)=\prod_{j=1,3}\frac{\sinh\vartheta+i\sin(ja)}{\sinh\vartheta-i\sin(ja)},\quad a=\pi/(2n)$}
The second <breather> is a bound pair of first <breathers> with constituent shifts $\pm ia$. The <bound-state fusion of factorized S-matrices> gives $S_{2,1}=S_{1,1}(\vartheta+ia)S_{1,1}(\vartheta-ia)$, which factors as displayed. The nearest physical-strip <pole> at $\vartheta=ia$ is a crossed-channel exchange of the first <breather>. At $n=3$ the more distant center <pole> is double, without changing that nearest-pole interpretation.
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