Voss surfaces in sine-Gordon hierarchies

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. A number of Voss nets are presented explicitly. As the main instrument, we reinterpret Guichard's transformations in terms of the recursion operators for sine-Gordon symmetries. Simultaneously, we clarify relevant aspects of Guthrie's formalism, paving the way for the future employment of the entire division algebra of recursion operators.