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Discrete & Computational Geometry

, Volume 56, Issue 2, pp 472–501 | Cite as

A \(2\times 2\) Lax Representation, Associated Family, and Bäcklund Transformation for Circular K-Nets

  • Tim Hoffmann
  • Andrew O. Sageman-Furnas
Article
  • 117 Downloads

Abstract

We present a \(2\times 2\) 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 associated families are given.

Keywords

Discrete differential geometry Discrete integrable systems Bäcklund transformations Multidimensional consistency 

Mathematics Subject Classification

53A05 37K25 37K35 

Notes

Acknowledgments

T.H. was supported by the DFG-Collaborative Research Center, TRR 109, “Discretization in Geometry and Dynamics.”

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Technische Universität MünchenGarchingGermany
  2. 2.Georg-August-Universität GöttingenGöttingenGermany

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