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.
- Group: B. Dynamics
- Principal Investigators: Prof. Dr. Yuri Suris, Prof. Dr. Alexander I. Bobenko
- Investigators: Dr. Matteo Petrera, Mats Vermeeren
- University: TU Berlin
- Term: since 2012
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
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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
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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
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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
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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
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external
arXiv
A construction of commuting systems of integrable symplectic birational maps
Authors:
Petrera, Matteo and
Suris, Yuri B
Note: Preprint
Date:
2016
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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
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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
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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
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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
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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
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arXiv
Two-dimensional variational systems on the root lattice $Q(A_N)$
Author:
Boll, R.
Note: preprint
Date:
2016
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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external
arXiv
Books
Mathematical Physics III - Integrable Systems of Classical Mechanics. Lecture Notes
Author:
Petrera, Matteo
Date:
2015
ISBN: 978-3-8325-3950-4
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Mathematical Physics II: Classical Statistical Mechanics. Lecture Notes
Author:
Petrera, Matteo
Date:
2014
ISBN: 978-3-8325-3719-7
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external
Mathematical Physics I: Dynamical Systems and Classical Mechanics. Lecture Notes
Author:
Petrera, Matteo
Date:
2013
ISBN: 978-3-8325-3569-8
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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