Explicit solutions from the Chebyshev Gauss–Mainardi–Codazzi zero-curvature representation

Construct explicit solutions of the Gauss–Mainardi–Codazzi equations under Chebyshev parameterisation by exploiting the zero-curvature representation obtained by Krasil'shchik and Marvan (1999) for the integrable classes identified there.

Background

An earlier work by Krasil'shchik and Marvan (1999) established a zero-curvature representation (ZCR) for integrable Gauss–Mainardi–Codazzi systems under Chebyshev parameterisation. While a ZCR is a standard gateway to exact solutions in soliton theory, deriving explicit solutions from it can be technically challenging.

In this paper, the authors report that they were not able to convert the ZCR into explicit solutions in general; they subsequently develop a geometric construction that yields solutions in the special case of concordant Chebyshev nets, leaving the broader task of extracting solutions from the ZCR in full generality unresolved within the text.