 Sine-Gordon Bäcklund transformation

For the angular field $u=\beta \varphi$ and coordinates $\xi =m(x+t)/2$, $\eta =m(x-t)/2$, let $w=(v+u)/2$ and $d=(v-u)/2$. The displayed relations imply $u_{\xi \eta }=\sin u$ and $v_{\xi \eta }=\sin v$. They define a Bäcklund transformation with the reciprocal parameter convention used in the Bianchi permutability for sine-Gordon Bäcklund transformations. Physical-field trigonometric arguments include the coupling $\beta$.