Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 075, 26 pages      arXiv:2511.15558      https://doi.org/10.3842/SIGMA.2026.075

Voss Surfaces in Sine-Gordon Hierarchies

Michal Marvan
Mathematical Institute in Opava, Silesian University in Opava, Na Rybnívcku 1, 746 01 Opava, Czech Republic

Received November 20, 2025, in final form July 23, 2026; Published online August 14, 2026

Abstract
We explore a method, initiated by Guichard in 1890, which allows to generate sequences of Voss surfaces, 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 operator for symmetries of the sine-Gordon equation and its inverse; 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, expanding the class of Voss surfaces obtainable by quadratures; 4) clarify relevant aspects of Guthrie's formalism, paving the way for the future employment of the entire division algebra of recursion operators. A number of Voss nets are presented explicitly.

Key words: Voss surface; Voss net; sine-Gordon equation; Moutard equation; recursion operator; Guichard transformation; pmKdV hierarchy; Khor'kova hierarchy; Guichard sequence; division algebra of recursion operators.

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