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-FurnasEmail author


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.


Discrete differential geometry Discrete integrable systems Bäcklund transformations Multidimensional consistency 

Mathematics Subject Classification

53A05 37K25 37K35 



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


  1. 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)CrossRefGoogle Scholar
  2. 2.
    Bobenko, A.I., Pinkall, U.: Discrete isothermic surfaces. J. Reine Angew. Math. 475, 187–208 (1996)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bobenko, A.I., Pinkall, U.: Discrete surfaces with constant negative Gaussian curvature and the Hirota equation. J. Differ. Geom. 43, 527–611 (1996)MathSciNetzbMATHGoogle Scholar
  4. 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. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bobenko, A.I., Suris, Y.B.: Integrable systems on quad-graphs. Int. Math. Res. Not. 2002(11), 573–611 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bobenko, A.I., Suris, Y.B.: Discrete Differential Geometry: Integrable Structure. Graduate Studies in Mathematics, vol. 98. American Mathematical Society, Providence (2008)Google Scholar
  8. 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)MathSciNetzbMATHGoogle Scholar
  9. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Doliwa, A., Nieszporski, M.: Darboux transformations for linear operators on two-dimensional regular lattices. J. Phys. A, Math. Theor. 42(45), 454001 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dorfmeister, J.F., Ivey, T., Sterling, I.: Symmetric pseudospherical surfaces I: general theory. Result. Math. 56(1–4), 3–21 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 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. 15.
    Hirota, R.: Nonlinear partial difference equations III: discrete sine-Gordon equation. J. Phys. Soc. Japan 43(6), 2079–2086 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 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. 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)CrossRefGoogle Scholar
  18. 18.
    Hoffmann, T., Sageman-Furnas, A.O., Wardetzky, M.: A discrete parametrized surface theory in \(\mathbb{R}^3\). arXiv:1412.7293v1, preprint (2014)
  19. 19.
    Konopelchenko, B.G., Schief, W.K.: Trapezoidal discrete surfaces: geometry and integrability. J. Geom. Phys. 31(2), 75–95 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 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)CrossRefGoogle Scholar
  22. 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)CrossRefzbMATHGoogle Scholar
  23. 23.
    Sauer, R.: Parallelogrammgitter als Modelle pseudosphärischer Flächen. Math. Z. 52(1), 611–622 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Schief, W.K.: On a maximum principle for minimal surfaces and their integrable discrete counterparts. J. Geom. Phys. 56(9), 1484–1495 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 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)CrossRefGoogle Scholar
  27. 27.
    Wunderlich, W.: Zur Differenzengeometrie der Flächen konstanter negativer Krümmung. Springer, Berlin (1951)zbMATHGoogle Scholar

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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