Topologically embedded pseudospherical cylinders

TL;DR

This paper classifies and explicitly describes all pseudospherical helicoids, solutions to the sine-Gordon equation with screw symmetry, including their topological types and invariants, solving a problem posed by Popov.

Contribution

It provides a complete explicit description of pseudospherical helicoids using elliptic functions and introduces a systematic method to classify pseudospherical twisted columns based on four invariants.

Findings

Explicit formulas for all pseudospherical helicoids in terms of elliptic functions.

Classification of pseudospherical helicoids into dense, immersed, or embedded cylinders.

Introduction of four invariants characterizing pseudospherical twisted columns.

Abstract

The class of traveling wave solutions of the sine-Gordon equation is known to be in 1-1 correspondence with the class of (necessarily singular) pseudospherical surfaces in Euclidean space with screw-motion symmetry: the pseudospherical helicoids. We explicitly describe all pseudospherical helicoids in terms of elliptic functions. This solves a problem posed by Popov [Lobachevsky geometry and modern nonlinear problems, Birkh\"auser/Springer, Cham, 2014]. As an application, countably many continuous families of topologically embedded pseudospherical helicoids are constructed. A (singular) pseudospherical helicoid is proved to be either a dense subset of a region bounded by two coaxial cylinders, a topologically immersed cylinder with helical self-intersections, or a topologically embedded cylinder with helical singularities, called for short a pseudospherical twisted column. Pseudospherical twisted columns are characterized by four phenomenological invariants: the helicity $\eta\in \mathbb{Z}_2$, the parity $\epsilon\in \mathbb{Z}_2$, the wave number $\mathfrak n\in \mathbb{N}$, and the aspect ratio $\mathfrak{d}>0$, up to translations along the screw axis. A systematic procedure for explicitly determining all pseudospherical twisted columns from the invariants is provided.