What is integrability of discrete variational systems? (Q2831316)

The notion of integrability is itself highly nontrivial, which is due to the absence of a comprehensive definition. However, in the case of discrete systems, one of the strongest integrability criteria is the notion of multidimensional consistency introduced in [\textit{F. W. Nijhoff} and \textit{A. J. Walker}, Glasg. Math. J. 43A, 109--123 (2001; Zbl 0990.39015); \textit{F. W. Nijhoff}, Phys. Lett. A 297, No. 1--2, 49--58 (2002; Zbl 0994.35105)] and, independently, in [\textit{A. I. Bobenko} and \textit{Y. B. Suris}, Int. Math. Res. Not. 2002, No. 11, 573--611 (2002; Zbl 1004.37053)]. As it is revealed in the title, this paper is devoted to the integrability of variational systems. In fact, the authors propose a notion of a pluri-Langrangian problem, as an analogue of the multidimensional consistency for variational systems. That is, they contribute into the development of the notion of multidimensional consistency by \textit{S. Lobb} and \textit{F. Nijhoff} [J. Phys. A, Math. Theor. 42, No. 45, Article ID 454013, 18 p. (2009; Zbl 1196.37117)], namely the Lagrangian formulation of discrete, multidimensionally consistent systems. In particular, Lobb and Nijhoff showed for a class of two-dimensional lattice (quad) equations that the associated Lagrangian two-forms (or multiforms for discrete integrable equations of more than two independent variables) are closed on solutions of the lattice equations. In this paper, the authors show that the Lagrangian two-forms are also closed on solutions of the so-called ``corner equations'', namely discrete Euler-Lagrange equations for all possible 3D-corners in \(\mathbb{Z}^m\). Moreover, they present an example of a pluri-Lagrangian system, which does not come from a multidimensionally consistent system of lattice equations.