Sine-Gordon two-kink solution ($u=4\arctan [v\sinh (m\Gamma x)/\cosh (m\Gamma vt)]$)

For $0<v<1$ and $\Gamma =(1-v^2{)}^{-1/2}$, this real two-soliton solution has angular winding $4\pi$. Its separated kink velocities are $\pm v$. At large times the centers satisfy $|m\Gamma x|=|m\Gamma vt|-\log v+o(1)$, giving a right-moving shift $-2\log v/(m\Gamma )$. It follows from Bianchi permutability for sine-Gordon Bäcklund transformations with oppositely signed seed parameters.