A 2x2 Lax representation, associated family, and B\"acklund transformation for circular K-nets

TL;DR

This paper introduces a 2x2 Lax representation for discrete circular nets with constant negative Gaussian curvature, enabling the derivation of Bäcklund transformations and associated families with explicit solutions.

Contribution

It provides a novel Lax representation linked to 4D consistency, leading to new Bäcklund transformations and explicit solutions for discrete circular nets.

Findings

Derived Bäcklund transformations for discrete circular nets.

Explicit solutions including Dini's surfaces and breather solutions.

All associated family members maintain constant Gaussian curvature.

Abstract

We present a 2x2 Lax representation for discrete circular nets of constant negative Gau{\ss} 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 B\"acklund transformations and an associated family. All the members of that family -- although no longer circular -- can be shown to have constant Gau{\ss} curvature as well. Explicit solutions for the B\"acklund transformations of the vacuum (in particular Dini's surfaces and breather solutions) and their respective associated families are given.