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.

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
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/