= Light-cone normalization of the sine-Gordon Bäcklund transformation
For $\tau=x+t$, $\rho=x-t$, the unit-mass <Sine-Gordon equation> is $\phi_{\tau\rho}=\tfrac14\sin\phi$. An auto-<Bäcklund transformation> with these coordinates is
$$
(\phi_1-\phi_0)_\rho=b\sin\frac{\phi_1+\phi_0}{2},\qquad
(\phi_1+\phi_0)_\tau=b^{-1}\sin\frac{\phi_1-\phi_0}{2}.
$$
Differentiate each relation in the other coordinate and add or subtract; the sine addition formula gives $\phi_{j,\tau\rho}=\tfrac14\sin\phi_j$ for $j=0,1$. Doubling both right-hand sides instead gives $\phi_{j,\tau\rho}=\sin\phi_j$, hence a sine-Gordon equation with mass squared four. Those doubled coefficients are appropriate for the half-scaled coordinates $(x+t)/2,(x-t)/2$.
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