A scattering theory of ultrarelativistic solitons

We construct a perturbative framework for understanding the collision of solitons (more precisely, solitary waves) in relativistic scalar field theories. Our perturbative framework is based on the suppression of the space-time interaction area proportional to $1/(\gamma v)$, where $v$ is the relative velocity of an incoming solitary wave and $\gamma = 1/\sqrt{1-v^2} \gg 1$. We calculate the leading order results for collisions of (1+1) dimensional kinks in periodic potentials, and provide explicit, closed form expressions for the phase shift and the velocity change after the collisions. We find excellent agreement between our results and detailed numerical simulations. Crucially, our perturbation series is controlled by a kinematic parameter, and hence not restricted to small deviations around integrable cases such as the Sine-Gordon model.