Inverse scattering and solitons in $A_{n-1}$ affine Toda field theories

TL;DR

This paper applies the inverse scattering method to $A_n$ affine Toda field theories, deriving soliton solutions and exploring additional oscillatory modes, while analyzing their properties and energy-momentum characteristics.

Contribution

It introduces a novel application of inverse scattering to these theories, revealing new solutions with oscillatory modes and linking energy-momentum to loop group extensions.

Findings

Recovered known single soliton solutions

Discovered solutions with non-linear oscillatory modes

Linked energy-momentum density to loop group central extension

Abstract

We implement the inverse scattering method in the case of the $A_n$ affine Toda field theories, by studying the space-time evolution of simple poles in the underlying loop group. We find the known single soliton solutions, as well as additional solutions with non-linear modes of oscillation around the standard solution, by studying the particularly simple case where the residue at the pole is a rank one projection. We show that these solutions with extra modes have the same mass and topological charges as the standard solutions, so we do not shed any light on the missing topological charge problem in these models. We also show that the integrated energy-momentum density can be calculated from the central extension of the loop group.