= Sine-Gordon Bäcklund transformation
{c}
{title2=$w_\xi=a^{-1}\sin d,\quad d_\eta=a\sin w$}
= Sine-Gordon Bäcklund transform
{c}
{synonym}
For the angular field $u=\beta\phi$ 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 $\beta$.
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