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. This is the goal of B02.
- Group: B. Dynamics
- Principal Investigators: Prof. Dr. Yuri Suris, Prof. Dr. Alexander I. Bobenko
- Investigators: Dr. Raphael Boll, Dr. Matteo Petrera, Mats Vermeeren
- University: TU Berlin
- Term: since 2012
Scientific Details+
The development of discrete differential geometry in recent years was intimately related to the development of discrete integrable systems. This is due to the fact that the majority (if not all) interesting special classes of surfaces and coordinate systems, smooth and discrete, turn out to be integrable, i.e., to be described via integrable systems. On the other hand a geometric interpretation provides us with new insights into the nature of integrability.
The main goal of this project will be a comprehensive analytical and structural study as well as a classification of multidimensional discrete integrable systems. The emphasis is on the geometric and algebraic structures behind integrability, such as multidimensional consistency, geometric characterization through incidence theorems of projective geometry, and algebraic relations for matrix minors.
Publications+
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
Papers
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
On the classification of multidimensionally consistent 3D maps
Authors:
Petrera, M. and
Suris, Y. B.
Note: preprint
Date:
Sep 2015
Download:
arXiv
Billiards in confocal quadrics as a pluri-Lagrangian system
Author:
Suris, Yuri B.
Note: preprint
Date:
2015
Download:
arXiv
Circle complexes and the discrete CKP equation
Authors:
Bobenko, A. I. and
Schief, W.
Date:
2015
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
On the construction of elliptic solutions of integrable birational maps
Authors:
Petrera, Matteo and
Pfadler, Andreas and
Suris, Yuri B.
Note: submitted to Experimental Mathematics
Date:
Sep 2014
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
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,
CaP
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. Raphael Boll +
Projects:
B02
University:
TU Berlin
E-Mail:
boll[at]math.tu-berlin.de
Dr. Matteo Petrera +
Projects:
B02
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