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 η∈Z_2 , the parity ϵ∈Z_2 , 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.

Topologically embedded helicoidal pseudospherical cylinders / Musso, Emilio; Nicolodi, Lorenzo. - In: NONLINEARITY. - ISSN 0951-7715. - 32:1(2019), pp. 22-45. [10.1088/1361-6544/aae720]

Topologically embedded helicoidal pseudospherical cylinders

Emilio Musso;Lorenzo Nicolodi
2019

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 η∈Z_2 , the parity ϵ∈Z_2 , 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.
Topologically embedded helicoidal pseudospherical cylinders / Musso, Emilio; Nicolodi, Lorenzo. - In: NONLINEARITY. - ISSN 0951-7715. - 32:1(2019), pp. 22-45. [10.1088/1361-6544/aae720]
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11381/2850833
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