= Solution
Only the empty binary configuration contributes to $f$, and only the occupied configuration contributes to $g$. Thus the <Hirota tau functions> give
$$
f=1,\qquad g=e^X,\qquad \phi=4\arctan e^X,\qquad X=\kappa x-\beta_1t+\gamma.
$$
Here $\beta_1$ denotes the velocity parameter, while $\beta$ without a subscript remains the coupling of <Sine-Gordon theory>. For $\kappa>0$, the field approaches the adjacent <scalar-field vacua> $0$ and $2\pi$ at the two ends of space, so its <topological charge> is $1$. Its center is $X=0$, giving
$$
\boxed{v=\frac{\beta_1}{\kappa},\qquad x_{\rm center}(t)=vt-\frac\gamma\kappa,\qquad |v|<1.}
$$
The constraint $\kappa^2-\beta_1^2=1$ implies $\kappa=(1-v^2)^{-1/2}$. Hence this is precisely a <Lorentz boost> of the static <Sine-Gordon kink>, with the expected <Lorentz contraction>. As a direct check, if $F(X)=4\arctan e^X$, then $F'=2\operatorname{sech}X$ and $F''=\sin F$, so $(\partial_t^2-\partial_x^2)\phi+\sin\phi=(\beta_1^2-\kappa^2)F''+\sin F=0$.
The real-parameter condition also permits $\kappa<0$. That choice reverses the <topological charge> and describes an <antikink>. The all-<kink> scattering formulas below use $\kappa_i>0$; the orientation dependence is stated explicitly at the end of the two-body calculation.
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