title: What is integrability of discrete variational systems?
abstract: We propose a notion of a pluri-Lagrangian problem, which should be understood as an analogue of multi-dimensional consistency for variational systems. This is a development along the line of research of discrete integrable Lagrangian systems initiated in 2009 by Lobb and Nijhoff, however, having its more remote roots in the theory of pluriharmonic functions, in theZ-invariant models of statistical mechanics and their quasiclassical limit, as well as in the theory of variational symmetries going back to Noether. Ad-dimensional pluri-Lagrangian problem can be described as follows: given ad-form [Formula: see text] on anm-dimensional space (called multi-time,m>d), whose coefficients depend on a sought-after functionxofmindependent variables (called field), find those fieldsxwhich deliver critical points to the action functionals [Formula: see text] foranyd-dimensional manifoldΣin the multi-time. We derive the main building blocks of the multi-time Euler-Lagrange equations for a discrete pluri-Lagrangian problem withd=2, the so-called corner equations, and discuss the notion of consistency of the system of corner equations. We analyse the system of corner equations for a special class of three-point two-forms, corresponding to integrable quad-equations of the ABS list. This allows us to close a conceptual gap of the work by Lobb and Nijhoff by showing that the corresponding two-forms are closed not only on solutions of (non-variational) quad-equations, but also on general solutions of the corresponding corner equations. We also find an example of a pluri-Lagrangian system not coming from a multi-dimensionally consistent system of quad-equations.
authors: Institut für Mathematik, MA 7-2 , Technische Universität Berlin , Str. des 17. Juni 136, 10623 Berlin, Germany.
authors: Raphael
authors: R
authors: Boll
authors: Institut für Mathematik, MA 7-2 , Technische Universität Berlin , Str. des 17. Juni 136, 10623 Berlin, Germany.
authors: Matteo
authors: M
authors: Petrera
authors: Institut für Mathematik, MA 7-2 , Technische Universität Berlin , Str. des 17. Juni 136, 10623 Berlin, Germany.
authors: Yuri B
authors: YB
authors: Suris
keywords: Euler–Lagrange equations; discrete forms; discrete integrable systems; multi-dimensional consistency; variational systems