A scattering theory of ultrarelativistic solitons

TL;DR

This paper develops a perturbative scattering theory for ultrarelativistic solitons in scalar field theories, providing explicit formulas for collision outcomes and validating them with numerical simulations.

Contribution

It introduces a velocity-based perturbative framework for soliton collisions that applies beyond integrable models, with explicit collision outcome formulas.

Findings

Explicit phase shift and velocity change formulas derived

Excellent agreement with numerical simulations demonstrated

Framework applicable to non-integrable relativistic scalar theories

Abstract

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.