A 2× Lax Representation, Associated Family, and Bäcklund Transformation for Circular K-Nets

We present a 2 × 2 Lax representation for discrete circular nets of constant negative Gauß curvature. It is tightly linked to the 4D consistency of the Lax representation of discrete K-nets (in asymptotic line parametrization). The description gives rise to Bäcklund transformations and an associated family. All the members of that family – although no longer circular – can be shown to have constant Gauß curvature as well. Explicit solutions for the Bäcklund transformations of the vacuum (in particular Dini’s surfaces and breather solutions) and their respective ∗T.H. was supported by the DFG-Collaborative Research Center, TRR 109, ”Discretization in Geometry and Dynamics.” †[email protected] ‡[email protected]