Letters in Mathematical Physics

, Volume 107, Issue 11, pp 2013–2027 | Cite as

On the classification of multidimensionally consistent 3D maps

  • Matteo Petrera
  • Yuri B. Suris


We classify multidimensionally consistent maps given by (formal or convergent) series of the following kind:
$$\begin{aligned} T_k x_{ij}=x_{ij} + \sum _{m=2}^\infty A_{ij ; \, k}^{(m)}(x_{ij},x_{ik},x_{jk}), \end{aligned}$$
where \(A_{ij;\, k}^{(m)}\) are homogeneous polynomials of degree m of their respective arguments. The result of our classification is that the only non-trivial multidimensionally consistent map in this class is given by the well-known symmetric discrete Darboux system
$$\begin{aligned} T_k x_{ij}=\frac{x_{ij}+x_{ik}x_{jk}}{\sqrt{1-x_{ik}^2}\sqrt{1-x_{jk}^2}}. \end{aligned}$$


Multidimensional consistency Darboux system Integrable three-dimensional maps 

Mathematics Subject Classification

37J35 39A12 39B12 51D25 



This research is supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics.” We thank the referees for their useful remarks which helped us to improve the presentation.


  1. 1.
    Adler, V.E., Bobenko, A.I., Suris, Yu.B.: Classification of integrable equations on quad-graphs. The consistency approach. Commun. Math. Phys. 223, 513–543 (2003)Google Scholar
  2. 2.
    Adler, V.E., Bobenko, A.I., Suris, Yu.B.: Discrete nonlinear hyperbolic equations. Classification of integrable cases. Funct. Anal. Appl. 43, 3–17 (2009)Google Scholar
  3. 3.
    Adler, V.E., Bobenko, A.I., Suris, Yu.B.: Classification of integrable discrete equations of octahedron type. Int. Math. Res. Not. 8, 1822–1889 (2012)Google Scholar
  4. 4.
    Akhmetshin, A.A., Volvovski, Yu.S., Krichever, I.M.: Discrete analogs of the Darboux–Egorov metrics. Proc. Steklov Inst. Math. 225, 16–39 (1999)Google Scholar
  5. 5.
    Bazhanov, V.V., Mangazeev, V.V., Smirnov, S.M.: Quantum geometry of three-dimensional lattices. J. Stat. Mech.: Theor. Exp. 7, P07004 (2008)MathSciNetGoogle Scholar
  6. 6.
    Bobenko, A.I., Suris, Yu.B.: Integrable systems on quad-graphs. Int. Math. Res. Not. 11, 573–611 (2002)Google Scholar
  7. 7.
    Bobenko, A.I., Suris, Yu.B.: Discrete Differential Geometry. Integrable Structure (Graduate Studies in Mathematics), vol. 98. AMS, Providence (2008)Google Scholar
  8. 8.
    Boll, R.: Classification of 3D consistent quad-equations. J. Nonlinear Math. Phys. 18, 337–365 (2011)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Doliwa, A.: The C-(symmetric) quadrilateral lattice, its transformations and the algebro-geometric construction. J. Geom. Phys. 60, 690–707 (2010)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Doliwa, A., Santini, P.M.: The symmetric, D-invariant and Egorov reductions of the quadrilateral lattice. J. Geom. Phys. 36, 60–102 (2000)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Kashaev, R.: On discrete three-dimensional equations associated with the local Yang–Baxter relation. Lett. Math. Phys. 38, 389–397 (1996)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    King, A.D., Schief, W.K.: Application of an incidence theorem for conics: Cauchy problem and integrability of the dCKP equation. J. Phys. A 39, 1899–1913 (2006)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Konopelchenko, B.G., Schief, W.K.: Three-dimensional integrable lattices in Euclidean spaces: conjugacy and orthogonality. Proc. R. Soc. Ser. A 454, 3075–3104 (1998)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Nijhoff, F.W.: Lax pair for Adler (lattice Krichever–Novikov) system. Phys. Lett. A 297, 49–58 (2002)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Nijhoff, F.W., Walker, A.: The discrete and continuous Painlevé VI hierarchy and the Garnier systems. Glasg. Math. J. A 43, 109–123 (2001)CrossRefzbMATHGoogle Scholar
  16. 16.
    Petrera, M., Suris, Yu.B.: Spherical geometry and integrable systems. Geom. Dedic. 169, 83–98 (2014)Google Scholar
  17. 17.
    Schief, W.K.: Lattice geometry of the discrete Darboux, KP, BKP and CKP equations. Menelaus’ and Carnot’s theorems. J. Nonlinear Math. Phys. 10(2), 194–208 (2003)ADSzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Institut für Mathematik, MA 7-1Technische Universität BerlinBerlinGermany

Personalised recommendations