Light-cone normalization of the sine-Gordon Bäcklund transformation

For $\tau =x+t$, $\rho =x-t$, the unit-mass Sine-Gordon equation is ${\varphi }_{\tau \rho }=\frac{1}{4}\sin \varphi$. An auto-Bäcklund transformation with these coordinates isDifferentiate each relation in the other coordinate and add or subtract; the sine addition formula gives ${\varphi }_{j,\tau \rho }=\frac{1}{4}\sin {\varphi }_j$ for $j=0,1$. Doubling both right-hand sides instead gives ${\varphi }_{j,\tau \rho }=\sin {\varphi }_j$, hence a sine-Gordon equation with mass squared four. Those doubled coefficients are appropriate for the half-scaled coordinates $(x+t)/2,(x-t)/2$.