 Solution

Choose the positive-exponential branches from part (i) and set their additive constants to zero. First take $0<b<1$, and defineThen $\tan ({\varphi }_b/4)=e^{\gamma (x-vt)}$ and $\tan ({\varphi }_a/4)=e^{-\gamma (x+vt)}$. The tangent subtraction formula givesConsequently the allowed Sine-Gordon superposition formula produces the smooth fieldThis is the negative of the Sine-Gordon two-kink solution, and hence a two-antikink configuration. The auxiliary seeds have opposite topological charges, but their charges cannot simply be added to infer the charge of the nonlinear two-step Bäcklund transformation. Indeed, the displayed final field tends to $2\pi$ at the left spatial end and $-2\pi$ at the right, so its total topological charge is $-2$.

Let $T=|t|$ become large. Near the right transition, $x=vT+O(1)$, the tangent argument has the asymptotic formso the local field is $-4\arctan e^{\gamma (x-vT)+\log v}$, a single antikink. Near the left transition, the local field is $4\arctan e^{-\gamma (x+vT)+\log v}+o(1)$, again a decreasing antikink. The resulting asymptotic center lines areThus two incoming antikinks with topological charges $(-1,-1)$ and velocities $(+v,-v)$ separate again with exactly the same topological charges and velocities. There is no radiative tail in these asymptotic profiles. Labeling the outgoing objects by their preserved rapidities makes this elastic soliton scattering; labeling the left and right lumps instead describes reflection with exchanged velocities.

For the right-moving soliton, its incoming intercept is ${\gamma }^{-1}\log v$ and its outgoing intercept is $-{\gamma }^{-1}\log v$. The spatial shifts are therefore $\Delta x_{+}=-2{\gamma }^{-1}\log v$ and $\Delta x_{-}=2{\gamma }^{-1}\log v$. Define the soliton time delay as the change in arrival time at a fixed distant spatial point relative to continuation of the incoming straight line, so $\Delta t=-\Delta x/v_{\mathrm{particle}}$. Both objects have the same signed soliton time delay, which is an advance:This is the Sine-Gordon two-kink time advance. In physical coordinates $T_{\mathrm{phys}}=t/m$, the time shift is $2\log v/(m\gamma v)$. The explicit intercepts fix the sign convention unambiguously.

The remaining real parameter choices are covered without changing the calculation. For any $b\ne 0$ with $b^2\ne 1$, put $v_b=(1-b^2)/(1+b^2)$, $u=|v_b|$, $\gamma =(|b|+|b{|}^{-1})/2$ and $ϵ=\mathrm{sgn}(b{v}_b)$. The same choice of zero additive constants givesEach scattered object's topological charge is $-ϵ$, the velocities are $\pm u$, and the signed soliton time delay is $2\log u/(\gamma u)$. If $b=\pm 1$, the superposition coefficient vanishes and this representative is the vacuum; there is no pair of separated moving solitons and no scattering delay to assign. Thus the scattering conclusion requires the nondegenerate case $0<u<1$.