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

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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.

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Notes

  1. For more general edge-bipartite graphs, vertices with valence greater than four might not have a continuous limit in the classical sense. For example, if the quad net is a discrete constant negative Gauß curvature surface parametrized by curvature lines, then these points are something like “Lorentz umbilics” [13].

  2. When \(\varepsilon = -2 \sin \delta \) and \(\phi = 2 \delta \) for some \(\delta \in \mathbbm {R}\) the squared radii (14) are constant and equal to \(\sin ^2\delta \) and the cK-net is trivial, formed from a rhombic K-net pseudosphere given in [3].

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Acknowledgments

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

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Authors and Affiliations

  1. Technische Universität München, Boltzmannstrasse 3, 85748, Garching, Germany

    Tim Hoffmann

  2. Georg-August-Universität Göttingen, Lotzestrasse 16-18, 37083, Göttingen, Germany

    Andrew O. Sageman-Furnas

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  1. Tim Hoffmann
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Correspondence to Andrew O. Sageman-Furnas.

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Hoffmann, T., Sageman-Furnas, A.O. A \(2\times 2\) Lax Representation, Associated Family, and Bäcklund Transformation for Circular K-Nets. Discrete Comput Geom 56, 472–501 (2016). https://doi.org/10.1007/s00454-016-9802-6

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