Skip to main content
SpringerLink
Log in

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

  • Published:
Discrete & Computational Geometry Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

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].

References

  1. Bobenko, A.I.: Surfaces in terms of 2 by 2 matrices: old and new integrable cases. In: Fordy, A.P., Wood, J.C. (eds.) Harmonic Maps and Integrable Systems, pp. 83–129. Vieweg, Braunschweig/Wiesbaden (1994)

    Chapter  Google Scholar 

  2. Bobenko, A.I., Pinkall, U.: Discrete isothermic surfaces. J. Reine Angew. Math. 475, 187–208 (1996)

    MathSciNet  MATH  Google Scholar 

  3. Bobenko, A.I., Pinkall, U.: Discrete surfaces with constant negative Gaussian curvature and the Hirota equation. J. Differ. Geom. 43, 527–611 (1996)

    MathSciNet  MATH  Google Scholar 

  4. Bobenko, A.I., Pinkall, U.: Discretization of surfaces and integrable systems. In: Bobenko, A.I., Seiler, R. (eds.) Discrete Integrable Geometry and Physics, pp. 3–58. Oxford University Press, Oxford (1999)

    Google Scholar 

  5. Bobenko, A.I., Pottmann, H., Wallner, J.: A curvature theory for discrete surfaces based on mesh parallelity. Math. Ann. 348(1), 1–24 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bobenko, A.I., Suris, Y.B.: Integrable systems on quad-graphs. Int. Math. Res. Not. 2002(11), 573–611 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bobenko, A.I., Suris, Y.B.: Discrete Differential Geometry: Integrable Structure. Graduate Studies in Mathematics, vol. 98. American Mathematical Society, Providence (2008)

  8. Cieśliński, J.: The spectral interpretation of \(n\)-spaces of constant negative curvature immersed in \(\mathbb{R}^{2n-1}\). Phys. Lett. A 236(5–6), 425–430 (1997)

    MathSciNet  MATH  Google Scholar 

  9. Cieśliński, J.L.: Pseudospherical surfaces on time scales: a geometric definition and the spectral approach. J. Phys. A, Math. Theor. 40(42), 12525–12538 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cieśliński, J., Doliwa, A., Santini, P.M.: The integrable discrete analogues of orthogonal coordinate systems are multi-dimensional circular lattices. Phys. Lett. A 235(5), 480–488 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Doliwa, A., Nieszporski, M.: Darboux transformations for linear operators on two-dimensional regular lattices. J. Phys. A, Math. Theor. 42(45), 454001 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Doliwa, A., Nieszporski, M., Santini, P.M.: Asymptotic lattices and their integrable reductions: I. The Bianchi–Ernst and the Fubini–Ragazzi lattices. J. Phys. A, Math. Gen. 34(48), 10–423 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dorfmeister, J.F., Ivey, T., Sterling, I.: Symmetric pseudospherical surfaces I: general theory. Result. Math. 56(1–4), 3–21 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hertrich-Jeromin, U., Hoffmann, T., Pinkall, U.: A discrete version of the Darboux transform for isothermic surfaces. In: Bobenko, A.I., Seiler, R. (eds.) Discrete Integrable Geometry and Physics, pp. 59–81. Oxford University Press, Oxford (1999)

    Google Scholar 

  15. Hirota, R.: Nonlinear partial difference equations III: discrete sine-Gordon equation. J. Phys. Soc. Japan 43(6), 2079–2086 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hoffmann, T.: Discrete Amsler surfaces and a discrete Painlevé III equation. In: Bobenko, A.I., Seiler, R. (eds.) Discrete Integrable Geometry and Physics, pp. 83–96. Oxford University Press, Oxford (1999)

    Google Scholar 

  17. Hoffmann, T.: Discrete Hashimoto surfaces and a doubly discrete smoke-ring flow. In: Bobenko, A.I., Sullivan, J.M., Schröder, P., Ziegler, G.M. (eds.) Discrete Differential Geometry, pp. 95–115. Springer, Berlin (2008)

    Chapter  Google Scholar 

  18. Hoffmann, T., Sageman-Furnas, A.O., Wardetzky, M.: A discrete parametrized surface theory in \(\mathbb{R}^3\). arXiv:1412.7293v1, preprint (2014)

  19. Konopelchenko, B.G., Schief, W.K.: Trapezoidal discrete surfaces: geometry and integrability. J. Geom. Phys. 31(2), 75–95 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nimmo, J.J.C., Schief, W.K.: Superposition principles associated with the Moutard transformation: an integrable discretization of a (2+1)-dimensional sine-Gordon system. Proc. R. Soc. A, Math. Phys. Eng. Sci. 453(1957), 255–279 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pinkall, U.: Designing cylinders with constant negative curvature. In: Bobenko, A.I., Schröder, P., Sullivan, J.M., Ziegler, G.M. (eds.) Discrete Differential Geometry, pp. 57–66. Springer, Berlin (2008)

    Chapter  Google Scholar 

  22. Rogers, C., Schief, W.K.: Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  23. Sauer, R.: Parallelogrammgitter als Modelle pseudosphärischer Flächen. Math. Z. 52(1), 611–622 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  24. Schief, W.K.: On the unification of classical and novel integrable surfaces. II. Difference geometry. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 459(2030), 373–391 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Schief, W.K.: On a maximum principle for minimal surfaces and their integrable discrete counterparts. J. Geom. Phys. 56(9), 1484–1495 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sym, A.: Soliton surfaces and their applications (soliton geometry from spectral problems). Geometric Aspects of the Einstein Equations and Integrable Systems. Lecture Notes in Physics, pp. 154–231. Springer, Berlin (1985)

    Chapter  Google Scholar 

  27. Wunderlich, W.: Zur Differenzengeometrie der Flächen konstanter negativer Krümmung. Springer, Berlin (1951)

    MATH  Google Scholar 

Download references

Acknowledgments

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

Author information

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

Authors
  1. Tim Hoffmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrew O. Sageman-Furnas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrew O. Sageman-Furnas.

Additional information

Editor in Charge: Günter M. Ziegler

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00454-016-9802-6

Keywords

Mathematics Subject Classification

Search

Navigation