 Soliton time delay

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.