B02
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.

Publications+

Papers
Commutativity in Lagrangian and Hamiltonian mechanics

Authors: Sridhar, Ananth and Suris, Yuri B
Note: Preprint
Date: 2018
Download: arXiv

A construction of a large family of commuting pairs of integrable symplectic birational four-dimensional maps

Authors: Petrera, Matteo and Suris, Yuri B
Journal: Proceedings of the Royal Society of London A: Mathematical, 473(2198)
Date: 2017
DOI: 10.1098/rspa.2016.0535
Download: external arXiv

New classes of quadratic vector fields admitting integral-preserving Kahan--Hirota--Kimura discretizations

Authors: Petrera, Matteo and Zander, René
Journal: Journal of Physics A: Mathematical and Theoretical, 50(20):205203
Date: 2017
DOI: 10.1088/1751-8121/aa6a0f
Download: external arXiv

On the classification of multidimensionally consistent 3D maps

Authors: Petrera, Matteo and Suris, Yuri B
Journal: Letters in Mathematical Physics, 107(11):2013--2027
Date: 2017
DOI: 10.1007/s11005-017-0976-5
Download: external arXiv

On the construction of elliptic solutions of integrable birational maps

Authors: Petrera, Matteo and Pfadler, Andreas and Suris, Yuri B
Journal: Experimental Mathematics, 26(3):324--341
Date: 2017
DOI: 10.1080/10586458.2016.1166354
Download: external arXiv

Variational symmetries and pluri-Lagrangian systems in classical mechanics

Authors: Petrera, Matteo and Suris, Yuri B
Journal: Journal of Nonlinear Mathematical Physics, 24(sup1):121--145
Date: 2017
DOI: 10.1080/14029251.2017.1418058
Download: external arXiv

A construction of commuting systems of integrable symplectic birational maps

Authors: Petrera, Matteo and Suris, Yuri B
Note: Preprint
Date: 2016
Download: arXiv

A construction of commuting systems of integrable symplectic birational maps. Lie-Poisson case

Authors: Petrera, Matteo and Suris, Yuri B
Note: Preprint
Date: 2016
Download: arXiv

Billiards in confocal quadrics as a pluri-Lagrangian system

Author: Suris, Yuri B.
Journal: Theoretical and Applied Mechanics, 43(2):221--228
Date: 2016
DOI: 10.2298/TAM160304008S
Download: external arXiv

Circle complexes and the discrete CKP equation

Authors: Bobenko, Alexander I and Schief, Wolfgang K
Journal: International Mathematics Research Notices, 2017(5):1504--1561
Date: 2016
DOI: 10.1093/imrn/rnw021
Download: external arXiv

On the Lagrangian structure of integrable hierarchies

Authors: Suris, Yu. B. and Vermeeren, M.
In Collection: Advances in Discrete Differential Geometry, Springer
Date: 2016
Download: arXiv

On the variational interpretation of the discrete KP equation

Authors: Boll, R. and Petrera, M. and Suris, Yu. B..
In Collection: Advances in Discrete Differential Geometry, Springer
Date: 2016
Download: arXiv

Two-dimensional variational systems on the root lattice $Q(A_N)$

Author: Boll, R.
Note: preprint
Date: 2016
Download: arXiv

Discrete line complexes and integrable evolution of minors

Authors: Bobenko, A. I. and Schief, W.
Journal: Proc. Royal Soc. A, 471(2175):23 pp.
Date: 2015
DOI: 10.1098/rspa.2014.0819
Download: external arXiv

Discrete pluriharmonic functions as solutions of linear pluri-Lagrangian systems

Authors: Bobenko, A. I. and Suris, Yu. B.
Journal: Commun. Math. Phys., 336(1):199--215
Date: 2015
Download: arXiv

On integrability of discrete variational systems: Octahedron relations

Authors: Boll, R. and Petrera, M. and Suris, Yu. B.
Journal: Internat. Math. Res. Notes, 2015:rnv140, 24 pp.
Date: 2015
Download: arXiv

Variational symmetries and pluri-Lagrangian systems

Author: Suris, Yu. B.
In Collection: Dynamical Systems, Number Theory and Applications: A Festschrift in Honor of Professor Armin Leutbecher's 80th Birthday, World Scientific
Date: 2015
Download: arXiv

Landau-Lifshitz equation, uniaxial anisotropy case: Theory of exact solutions

Authors: Bikbaev, R.F. and Bobenko, A.I. and Its, A.R.
Journal: Theoretical and Mathematical Physics, 178(2):143-193
Date: Feb 2014
DOI: 10.1007/s11232-014-0135-4
Download: external

What is integrability of discrete variational systems?

Authors: Boll, Raphael and Petrera, Matteo and Suris, Yuri B.
Journal: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 470(2162)
Date: Feb 2014
DOI: 10.1098/rspa.2013.0550
Download: external arXiv

On Weingarten Transformations of Hyperbolic Nets

Authors: Huhnen-Venedey, Emanuel and Schief, Wolfgang K.
Journal: International Mathematics Research Notices
Date: 2014
DOI: 10.1093/imrn/rnt354
Download: external arXiv

On Bianchi permutability of Bäcklund transformations for asymmetric quad-equations

Author: Boll, Raphael
Journal: Journal of Nonlinear Mathematical Physics, 20(4):577-605
Date: Dec 2013
DOI: 10.1080/14029251.2013.865829
Download: external arXiv

Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Relativistic Toda-type systems

Authors: Boll, R. and Petrera, M. and Suris, Yu. B.
Journal: J. Phys. A: Math. Theor., 46(27):275024, 26 pp.
Date: 2013
DOI: 10.1088/1751-8113/46/27/275204
Download: external arXiv

Variational formulation of commuting Hamiltonian flows: multi-time Lagrangian 1-forms

Author: Suris, Yu. B.
Journal: J. Geometric Mechanics, 5(3):365--379
Date: 2013
DOI: 10.3934/jgm.2013.5.365
Download: external arXiv

On Discrete Integrable Equations with Convex Variational Principles

Authors: Bobenko, Alexander I. and Günther, Felix
Journal: Letters in Mathematical Physics, 102(2):181-202
Date: Sep 2012
DOI: 10.1007/s11005-012-0583-4
Download: external arXiv

S. Kovalevskaya system, its generalization and discretization

Authors: Petrera, M. and Suris, Y. B.
Journal: Frontiers of Mathematics in China, 2013, 8, No. 5, p. 1047-1065
Date: Aug 2012
DOI: 10.1007/s11464-013-0305-y
Download: external arXiv

Spherical geometry and integrable systems

Authors: Petrera, M. and Suris, Y. B.
Journal: Geometriae Dedicata
Date: Aug 2012
DOI: 10.1007/s10711-013-9843-4
Download: external arXiv


Books
Mathematical Physics III - Integrable Systems of Classical Mechanics. Lecture Notes

Author: Petrera, Matteo
Date: 2015
ISBN: 978-3-8325-3950-4
Download: external

Mathematical Physics II: Classical Statistical Mechanics. Lecture Notes

Author: Petrera, Matteo
Date: 2014
ISBN: 978-3-8325-3719-7
Download: external

Mathematical Physics I: Dynamical Systems and Classical Mechanics. Lecture Notes

Author: Petrera, Matteo
Date: 2013
ISBN: 978-3-8325-3569-8
Download: external


Team+

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: Strasse des 17. Juni 136, 10623 Berlin, Germany
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/


Mats Vermeeren   +

Projects: B02
University: TU Berlin
E-Mail: vermeer[at]math.tu-berlin.de