Notes on Periodic Solitons

TL;DR

This paper analyzes static solutions of the sine-Gordon model on a cylinder, exploring their classification, energy, and stability using elliptic functions, modular transformations, and supersymmetric quantum mechanics.

Contribution

It provides a detailed review of periodic and quasi-periodic sine-Gordon solutions, their relation via modular transformations, and examines their stability through supersymmetric quantum mechanics.

Findings

Solutions can be decomposed into kinks and anti-kinks or monotonic kink trains.

Energy expressions are derived using elliptic functions and contour integrals.

Unstable configurations are linked to singular superpotentials, indicating negative fluctuation modes.

Abstract

We consider static solutions of the sine-Gordon theory defined on a cylinder, which can be either periodic or quasi-periodic in space. They are described by the different modes of a simple pendulum moving in an inverted effective potential and correspond to its libration or rotation. We review the decomposition of the solutions into an oscillatory sum of alternating kinks and anti-kinks or into a monotonic train of kinks, respectively, using properties of elliptic functions. The two sectors are naturally related to each other by a modular transformation, whereas the underlying spectral curve of the model can be used to express the energy of the static configurations in terms of contour integrals \`a la Seiberg-Witten in either case. The stability properties are also examined by means of supersymmetric quantum mechanics, where we find that the unstable configurations are associated to singular superpotentials, thus allowing for negative modes in the spectrum of small fluctuations.