# Salient Features of Dressed Elliptic String Solutions on $\mathbb{R}\times$S$^2$

## TL;DR

This paper explores the physical properties, classifications, and stability of dressed elliptic string solutions on  imes Sb2, revealing interactions, topological features, and differences between solution classes with potential implications for dual field theories.

## Contribution

It provides a detailed analysis of dressed elliptic strings, classifies solutions, examines their interactions and stability, and compares their properties to the sine-Gordon model.

## Key findings

- Identification of two classes of solutions: kinks and periodic disturbances.
- Discovery of four subclasses of closed dressed strings with spike interactions.
- Observation of instabilities related to superluminal kinks on elliptic solutions.

## Abstract

We analyse several physical aspects of the dressed elliptic strings propagating on $\mathbb{R} \times \mathrm{S}^2$ and of their counterparts in the Pohlmeyer reduced theory, i.e. the sine-Gordon equation. The solutions are divided into two wide classes; kinks which propagate on top of elliptic backgrounds and those which are non-localised periodic disturbances of the latter. The former class of solutions obey a specific equation of state that is in principle experimentally verifiable in systems which realize the sine-Gordon equation. Among both of these classes, there appears to be a particular class of interest the closed dressed strings. They in turn form four distinct subclasses of solutions. Unlike the closed elliptic strings, these four subclasses, exhibit interactions among their spikes. These interactions preserve a carefully defined turning number, which can be associated to the topological charge of the sine-Gordon counterpart. One particular class of those closed dressed strings realizes instabilities of the seed elliptic solutions. The existence of such solutions depends on whether a superluminal kink with a specific velocity can propagate on the corresponding elliptic sine-Gordon solution. Finally, the dispersion relations of the dressed strings are studied. A qualitative difference between the two wide classes of dressed strings is discovered. This would be an interesting subject for investigation in the dual field theory.