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An auxiliary function used in Hirota's bilinear method to encode a solution of an integrable partial differential equation. Products and ratios of tau functions replace nonlinear field variables; their differential equations become bilinear. In a Sine-Gordon multisoliton tau representation, two real tau functions encode the field as a continuous $4\arg (f+ig)$.

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.

Write the rapidity parameters as ${\kappa }_i=\cosh {\theta }_i$, ${\beta }_i=\sinh {\theta }_i$, and define $E_i=e^{X_i}$. The signed coefficient in the Sine-Gordon multisoliton tau representation isFor distinct rapidities, $0<{\alpha }_{12}<1$. In particular, $B_{12}$ cannot be taken as a real logarithm of a positive coefficient. The finite sums defining the Hirota tau functions can instead be evaluated directly with the real, negative $a_{12}$. They giveThe physical field is a continuous branch of a multivalued function, equivalently $\varphi =4\arg (f+ig)$ with the argument followed continuously. The principal inverse tangent alone jumps when $f$ changes sign.

Follow the first kink with $x=v_{1}t+O(1)$. Then $X_1=O(1)$ and $X_2={\kappa }_2(v_1-v_2)t+O(1)$. The two possible local limits arewhere the second field is written on the continuous kink branch. Thus both limits are single Sine-Gordon kinks of the same width and velocity, but their centers obey $X_1=0$ or $X_1+\log {\alpha }_{12}=0$. Following the second kink gives the same conclusion with labels exchanged. The incoming and outgoing velocities are thereforeThere is no change in the asymptotic rapidities or kink profiles.

Define the spatial shift as the outgoing center intercept minus the incoming center intercept. Since the large-$E_2$ limit occurs afterwards when $v_1>v_2$, and beforehand when $v_1<v_2$, the soliton time delay isThe time formula uses $\Delta t=-\Delta x/v_1$ and requires $v_1\ne 0$. Its dependence on the velocities is explicit on substitutingFor a faster right-moving kink, $\Delta x_{1|2}>0$ and $\Delta t<0$: it arrives earlier than its freely continued incoming trajectory. If $v_1=0$, report the finite spatial shift; a fixed-position arrival-time delay for a stationary kink is undefined. Coincident velocities are excluded from a separated collision asymptotic.

For completeness, allowing antikinks means ${\kappa }_i={\sigma }_i\cosh {\theta }_i$, ${\beta }_i={\sigma }_i\sinh {\theta }_i$, with ${\sigma }_i=\pm 1$. The velocities remain $\tanh {\theta }_i$. For opposite orientations, $|a_{12}|={\coth }^2[({\theta }_1-{\theta }_2)/2]$, and the general spatial shift isThis follows from the same two local limits; it makes explicit the orientation hypothesis behind the velocity-only all-kink answer.