Letters in Mathematical Physics

, Volume 102, Issue 2, pp 181–202 | Cite as

On Discrete Integrable Equations with Convex Variational Principles

  • Alexander I. Bobenko
  • Felix Günther


The Lagrangian structure of two-dimensional integrable systems on quad-graphs is investigated. We give reality conditions under which the action functionals are strictly convex. In particular, this gives uniqueness of solutions of Dirichlet boundary value problems. In some cases, we discuss also the existence of solutions. The integrability of combinatorial data is studied. In addition, a connection between (Q3) and circle patterns is discussed.

Mathematics Subject Classification (2010)

37J35 52C26 70S05 


discrete integrable systems integrable quad-equations Lagrangian formalism variational principle Dirichlet boundary value problem circle patterns 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adler V.E., Bobenko A.I., Suris Yu.B.: Classification of integrable equations on quad-graphs. The consistency approach. Commun. Math. Phys. 233(3), 513–543 (2003)MathSciNetADSzbMATHGoogle Scholar
  2. 2.
    Bazhanov V.V., Mangazeev V.V., Sergeev S.M.: Faddeev-Volkov solution of the Yang–Baxter equation and discrete conformal symmetry. Nucl. Phys. B 784(3), 234–258 (2007)MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 3.
    Bazhanov, V.V., Sergeev, S.M.: A master solution of the quantum Yang–Baxter equation and classical discrete integrable equations. arXiv:1006.0651 (2010)Google Scholar
  4. 4.
    Bazhanov V.V., Sergeev S.M.: Elliptic gamma-function and multi-spin solutions of the Yang–Baxter equation. Nucl. Phys. B 856(2), 475–496 (2012)MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Bobenko, A.I., Günther, F.: On discrete integrable equations with convex variational principles. arXiv:1111.6273 (2011)Google Scholar
  6. 6.
    Bobenko A.I., Springborn B.A.: Variational principles for circle patterns and Koebe’s theorem. Trans. Am. Math. Soc. 356(2), 659–689 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bobenko A.I., Suris Yu.B.: Integrable systems on quad-graphs. Int. Math. Res. Not. 2002(11), 573–611 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bobenko, A.I., Suris, Yu.B.: Discrete differential geometry: Integrable structures. In: Graduate Studies in Mathematics, vol. 98. AMS, Providence (2008)Google Scholar
  9. 9.
    Bobenko A.I., Suris Yu.B.: On the Lagrangian structure of integrable quad-equations. Lett. Math. Phys. 92(3), 17–31 (2010)MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. 10.
    Hietarinta J.: Searching for CAC-maps. J. Nonlinear Math. Phys. 12(Supplement 2), 223–230 (2005)MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Kenyon R., Schlenker J.-M.: Rhombic embeddings of planar quad-graphs. Trans. Am. Math. Soc. 357(9), 3443–3458 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Lobb S., Nijhoff F.W.: Lagrangian multiforms and multidimensional consistency. J. Phys. A: Math. Theor. 42(45), 454013 (2009)MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    Marsden J.E., Patrick G.W., Shkoller S.: Multisymplectic geometry, variational integrators and nonlinear PDEs. Commun. Math. Phys. 199(2), 351–395 (1998)MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. 14.
    Marsden J.E., Wendlandt J.M.: Mechanical integrators derived from a discrete variational principle. Physica D 106(3–4), 223–246 (1997)MathSciNetADSzbMATHGoogle Scholar
  15. 15.
    Moser J., Veselov A.P.: Discrete versions of some classical integrable systems and factorization of matrix polynomials. Commun. Math. Phys. 139(2), 217–243 (1991)MathSciNetADSzbMATHCrossRefGoogle Scholar
  16. 16.
    Nijhoff F.W.: Lax pair for the Adler (lattice Krichever-Novikov) system. Phys. Lett. A 297(1–2), 49–58 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  17. 17.
    Suris Yu.B.: Discrete Lagrangian models. In: Grammaticos, B., Kosmann-Schwarzbach, Y., Tamizhmani, T. (eds.) Lecture Notes in Physics, vol. 644, pp. 111–184. Springer, Berlin (2004)Google Scholar

Copyright information

© Springer 2012

Authors and Affiliations

  1. 1.Institut für Mathematik, MA 8-3Technische Universität BerlinBerlinGermany

Personalised recommendations