Sine-Gordon multisoliton tau representation ($\varphi =4\arg (f+ig)$)

For real parameters with ${\kappa }_i^2-b_i^2=1$, set $E_i=e^{{\kappa }_{i}x-b_{i}t+{\gamma }_i}$ and $a_{ij}=[({\kappa }_i-{\kappa }_j{)}^2-(b_i-b_j{)}^2]/[({\kappa }_i+{\kappa }_j{)}^2-(b_i+b_j{)}^2]$. Sum ${\prod }_i{E}_i^{{\mu }_i}{\prod }_{i<j}{a}_{ij}^{{\mu }_i{\mu }_j}$ over binary vectors of even parity for $f$ and odd parity for $g$. The Sine-Gordon equation solution is the continuous field $4\arg (f+ig)$. In the all-kink sector, ${\kappa }_i=\cosh {\theta }_i$, $b_i=\sinh {\theta }_i$ and $a_{ij}=-{\tanh }^2[({\theta }_i-{\theta }_j)/2]$. Treating these coefficients as positive would change the solution. Distinct rapidities give separated incoming and outgoing solitons.