Discrete Multidimensional Integrable Systems

Classifying and Structuring Multidimensional Discrete Integrable Systems

In recent years, there has been great interest among differential geometers for discovered discrete integrable systems, since many discrete systems with important applications turned out to be integrable. However, there is still no exhaustive classification of these systems, in particular concerning their geometric and combinatoric structure. Here, we investigate and classify multidimensional discrete integrable systems.

Scientific Details+

The project aims at the study and classification of multidimensional discrete integrable systems. It is now an outgrowth of two projects from the first funding period of CRC: B02 “Discrete multidimensional integrable systems” and B07 “Lagrangian multiform structure and multisymplectic discrete systems”. As such, it unifies different methodological approaches of those two projects to discrete integrable systems, namely, on the one hand, the emphasis on the combinatorial issues of the underlying lattice and the geometric interpretation of integrable systems, and, on the other hand, their variational (Lagrangian) interpretation.

During the first funding period we identified the M-system as a master system unifying a great variety of different 3D systems on the cubic lattice and developed a variational approach to integrable systems based on the notion of the pluri-Lagrangian structure. Now, we intend to identify discrete integrable master equations on lattices different from Zm, playing the role similar to that of the so-called M-system (governing the integrable evolution of minors of arbitrary matrices). In addition, we want to classify discrete forms generating pluri-Lagrangian systems.This should serve as a preparation to finding the pluri-Lagrangian structure of the fundamental 3D integrable systems of cubic type, including the discrete BKP equation and its Schwarzian version, and after that to the classification of discrete 3- and 4-forms on different lattices generating pluri-Lagrangian systems. Combining both approaches, we hope to finally obtain the solution of a long-standing classification problem concerning integrable systems: why 2D integrable systems are abundant, while only a few integrable 3D systems and no integrable 4D systems are known.


  • Mats Vermeeren.
    Modified equations for variational integrators applied to Lagrangians linear in velocities.
    J. Geom. Mech., 11(1):1–22, March 2019.
    arXiv:1709.09567, doi:10.3934/jgm.2019001.
  • Mats Vermeeren.
    Continuum limits of pluri-Lagrangian systems.
    J. Integrable Syst., 4(1):1–34, February 2019.
    arXiv:1706.06830, doi:10.1093/integr/xyy020.
  • Mats Vermeeren.
    A variational perspective on continuum limits of ABS and lattice GD equations.
    SIGMA Symmetry Integrability Geom. Methods Appl., 15:044, 2019.
  • Ananth Sridhar and Yuri B Suris.
    Commutativity in Lagrangian and Hamiltonian mechanics.
    J. Geom. Phys., 137:154–161, 2019.
    arXiv:1801.06076, doi:10.1016/j.geomphys.2018.09.019.
  • Mats Vermeeren, Alessandro Bravetti, and Marcello Seri.
    Contact variational integrators.
    Journal of Physics A: Mathematical and Theoretical, 52(44):445206, 2019.
  • Alexander Bobenko and Yuri Suris.
    Integrable linear systems on quad-graphs.
    preprint, 2019.
  • Matteo Petrera and Mats Vermeeren.
    Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs.
    preprint, 2019.
  • Mats Vermeeren.
    Numerical precession in variational discretizations of the Kepler problem.
    In K. Ebrahimi-Fard and M. Barbero Linan, editors, Discrete Mechanics, Geometric Integration and Lie–Butcher Series, pages 333–348. Springer, Cham, 2018.
  • Matteo Petrera and Yuri B Suris.
    A construction of a large family of commuting pairs of integrable symplectic birational four-dimensional maps.
    Proceedings of the Royal Society of London A: Mathematical, 2017.
    arXiv:1606.08238, doi:10.1098/rspa.2016.0535.
  • Mats Vermeeren.
    Modified equations for variational integrators.
    Num. Math., 137:1001–1037, 2017.
    arXiv:1505.05411, doi:10.1007/s00211-017-0896-4.
  • Matteo Petrera and René Zander.
    New classes of quadratic vector fields admitting integral-preserving Kahan–Hirota–Kimura discretizations.
    Journal of Physics A: Mathematical and Theoretical, 50(20):205203, 2017.
    arXiv:1610.03664, doi:10.1088/1751-8121/aa6a0f.
  • Matteo Petrera and Yuri B Suris.
    On the classification of multidimensionally consistent 3D maps.
    Letters in Mathematical Physics, 107(11):2013–2027, 2017.
    arXiv:1509.03129, doi:10.1007/s11005-017-0976-5.
  • Matteo Petrera, Andreas Pfadler, and Yuri B Suris.
    On the construction of elliptic solutions of integrable birational maps.
    Experimental Mathematics, 26(3):324–341, 2017.
    arXiv:1409.1741, doi:10.1080/10586458.2016.1166354.
  • Matteo Petrera and Yuri B Suris.
    Variational symmetries and pluri-Lagrangian systems in classical mechanics.
    Journal of Nonlinear Mathematical Physics, 24(sup1):121–145, 2017.
    arXiv:1710.01526, doi:10.1080/14029251.2017.1418058.
  • Matteo Petrera and Yuri B Suris.
    A construction of commuting systems of integrable symplectic birational maps.
    Preprint, 2016.
  • Matteo Petrera and Yuri B Suris.
    A construction of commuting systems of integrable symplectic birational maps. Lie-Poisson case.
    Preprint, 2016.
  • Yuri B. Suris.
    Billiards in confocal quadrics as a pluri-Lagrangian system.
    Theoretical and Applied Mechanics, 43(2):221–228, 2016.
    arXiv:1511.06123, doi:10.2298/TAM160304008S.
  • Alexander I Bobenko and Wolfgang K Schief.
    Circle complexes and the discrete CKP equation.
    International Mathematics Research Notices, 2017(5):1504–1561, 2016.
    arXiv:1509.04109, doi:10.1093/imrn/rnw021.
  • Yu. B. Suris and M. Vermeeren.
    On the Lagrangian structure of integrable hierarchies.
    In A.I. Bobenko, editor, Advances in Discrete Differential Geometry. Springer, Berlin-Heidelberg-New York, 2016.
  • R. Boll, M. Petrera, and Yu. B.. Suris.
    On the variational interpretation of the discrete KP equation.
    In A.I. Bobenko, editor, Advances in Discrete Differential Geometry. Springer, Berlin-Heidelberg-New York, 2016.
  • R. Boll.
    Two-dimensional variational systems on the root lattice $Q(A_N)$.
    preprint, 2016.
  • A. I. Bobenko and W. Schief.
    Discrete line complexes and integrable evolution of minors.
    Proc. Royal Soc. A, 471(2175):23 pp., 2015.
    arXiv:1410.5794, doi:10.1098/rspa.2014.0819.
  • A. I. Bobenko and Yu. B. Suris.
    Discrete pluriharmonic functions as solutions of linear pluri-Lagrangian systems.
    Commun. Math. Phys., 336(1):199–215, 2015.
  • R. Boll, M. Petrera, and Yu. B. Suris.
    On integrability of discrete variational systems: Octahedron relations.
    Internat. Math. Res. Notes, 2015:rnv140, 24 pp., 2015.
  • Yu. B. Suris.
    Variational symmetries and pluri-Lagrangian systems.
    In Th. Hagen, F. Rupp, and J. Scheurle, editors, Dynamical Systems, Number Theory and Applications: A Festschrift in Honor of Professor Armin Leutbecher's 80th Birthday. World Scientific, Singapore, 2015.
  • R.F. Bikbaev, A.I. Bobenko, and A.R. Its.
    Landau-Lifshitz equation, uniaxial anisotropy case: Theory of exact solutions.
    Theoretical and Mathematical Physics, 178(2):143–193, February 2014.
  • Raphael Boll, Matteo Petrera, and Yuri B. Suris.
    What is integrability of discrete variational systems?
    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, February 2014.
    URL: http://rspa.royalsocietypublishing.org/content/470/2162/20130550.abstract, arXiv:1307.0523, doi:10.1098/rspa.2013.0550.
  • Emanuel Huhnen-Venedey and Wolfgang K. Schief.
    On Weingarten Transformations of Hyperbolic Nets.
    International Mathematics Research Notices, 2014.
    URL: http://imrn.oxfordjournals.org/content/early/2014/01/24/imrn.rnt354.abstract, arXiv:1305.4783, doi:10.1093/imrn/rnt354.
  • Raphael Boll.
    On Bianchi permutability of Bäcklund transformations for asymmetric quad-equations.
    Journal of Nonlinear Mathematical Physics, 20(4):577–605, December 2013.
    arXiv:1211.4374, doi:10.1080/14029251.2013.865829.
  • R. Boll, M. Petrera, and Yu. B. Suris.
    Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Relativistic Toda-type systems.
    J. Phys. A: Math. Theor., 46(27):275024, 26 pp., 2013.
    arXiv:1408.2405, doi:10.1088/1751-8113/46/27/275204.
  • Yu. B. Suris.
    Variational formulation of commuting Hamiltonian flows: multi-time Lagrangian 1-forms.
    J. Geometric Mechanics, 5(3):365–379, 2013.
    arXiv:1212.3314, doi:10.3934/jgm.2013.5.365.
  • Alexander I. Bobenko and Felix Günther.
    On Discrete Integrable Equations with Convex Variational Principles.
    Letters in Mathematical Physics, 102(2):181–202, September 2012.
    arXiv:1111.6273, doi:10.1007/s11005-012-0583-4.
  • M. Petrera and Y. B. Suris.
    S. Kovalevskaya system, its generalization and discretization.
    Frontiers of Mathematics in China, 2013, 8, No. 5, p. 1047-1065, August 2012.
    arXiv:1208.3726, doi:10.1007/s11464-013-0305-y.
  • M. Petrera and Y. B. Suris.
    Spherical geometry and integrable systems.
    Geometriae Dedicata, August 2012.
    arXiv:1208.3625, doi:10.1007/s10711-013-9843-4.



Prof. Dr. Yuri Suris   +

Projects: B02, B10
University: TU Berlin
E-Mail: suris[at]math.tu-berlin.de
Website: http://page.math.tu-berlin.de/~suris/

Prof. Dr. Alexander I. Bobenko   +

Projects: A01, A02, C01, B02, Z, CaP, II
University: TU Berlin, Institut für Mathematik, MA 881
Address: MA 881
Tel: +49 (30) 314 24655
E-Mail: bobenko[at]math.tu-berlin.de
Website: http://page.math.tu-berlin.de/~bobenko/

Dr. Matteo Petrera   +

Projects: B02, Z
University: TU Berlin
E-Mail: petrera[at]math.tu-berlin.de
Website: http://page.math.tu-berlin.de/~petrera/