Voss surfaces in sine-Gordon hierarchies

TL;DR

This paper links classical transformations for pseudospherical surfaces to modern recursion operators in sine-Gordon theory, enabling systematic generation of Voss surfaces through quadratures and expanding the known classes of such surfaces.

Contribution

It identifies Guichard transformations with sine-Gordon recursion operators and introduces extended inverted operators to generate more Voss surfaces.

Findings

Guichard transformations are equivalent to sine-Gordon recursion operators

Length of Voss sequences can be derived from initial solution invariance

Extended operators increase the class of obtainable Voss surfaces

Abstract

We explore a method, initiated by Guichard in 1890, which allows to generate sequences of Voss surfaces by quadratures, starting from an arbitrarily chosen pseudospherical surface and a seed solution of the Moutard equation, by means of two simple transformations. In this paper we 1) identify the Guichard transformations with the well-known recursion operators for symmetries of the sine-Gordon equation; 2) prove a lemma which allows us to derive the length of Guichard's sequences from the invariance properties of the initial sine-Gordon solution; 3) introduce an extended class of inverted operators, increasing the number of Voss surfaces obtainable by quadratures. Several Voss nets are presented explicitly.