Soliton time delay ($\Delta t=-\Delta x/v$)

If a labeled soliton approaches $x=vt+a_{-}$ before a collision and $x=vt+a_{+}$ afterwards, its spatial shift is $\Delta x=a_{+}-a_{-}$. For $v\ne 0$, define its time delay at a fixed spatial point by $\Delta t=-\Delta x/v$. Thus a positive forward shift of a right-moving soliton is a negative time delay. For a stationary soliton, the spatial shift is well defined but this time-delay definition is not.

A many-soliton spatial shift is pairwise additive when $\Delta x_i={\sum }_{j\ne i}\Delta x_{i|j}$. In the all-kink sector of Sine-Gordon theory, $\Delta x_{i|j}=-\mathrm{sgn}(v_i-v_j)\log {\tanh }^2[({\theta }_i-{\theta }_j)/2]/\cosh {\theta }_i$. Dominant spectator exponentials in the Sine-Gordon multisoliton tau representation multiply the effective exponential of kink $i$, so their logarithms add. This is a classical signature of factorized scattering, with no independent many-body contribution to the asymptotic shift.