Topologically embedded helicoidal pseudospherical cylinders

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 (2014 Lobachevsky Geometry and Modern Nonlinear Problems (Berlin: Springer)). 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 η ∈ Z2, the parity ε ∈ Z2, the wave number n ∈ N, and the aspect ratio d > 0, up to translations along the screw axis. A systematic procedure for explicitly determining all pseudospherical twisted columns from the invariants is provided.