What is integrability of discrete variational systems?

Abstract

We propose a notion of a pluri-Lagrangian problem, which should be understood as an analog 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 the Z-invariant models of statistical mechanics and their quasiclassical limit, as well as in the theory of variational symmetries going back to Noether. A d-dimensional pluri-Lagrangian problem can be described as follows: given a d-form L on an m-dimensional space (called multi-time, m>d), whose coefficients depend on a sought-after function x of m independent variables (called field), find those fields x which deliver critical points to the action functionals $S_{\Sigma }={\int }_{\Sigma }L$ for any d-dimensional manifold $\Sigma$ in the multi-time. We derive the main building blocks of the multi-time Euler-Lagrange equations for a discrete pluri-Lagrangian problem with d=2, the so called corner equations, and discuss the notion of consistency of the system of corner equations. We analyze the system of corner equations for a special class of three-point 2-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 2-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 multidimensionally consistent system of quad-equations.

19 pp

References in corpus (3)

• Faddeev-Volkov solution of the Yang-Baxter Equation and Discrete Conformal Symmetry
• Variational formulation of commuting Hamiltonian flows: multi-time Lagrangian 1-forms
• Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Toda-type systems

Cited by in corpus (21)

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• Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs
• Discrete pluriharmonic functions as solutions of linear pluri-Lagrangian systems
• Continuum limits of pluri-Lagrangian systems
• Discrete time Toda systems
• Lagrangian 3-form structure for the Darboux system and the KP hierarchy
• Multiform description of the AKNS hierarchy and classical r-matrix
• Hamiltonian multiform description of an integrable hierarchy
• A variational perspective on continuum limits of ABS and lattice GD equations
• Extended Z-invariance for integrable vector and face models and multi-component integrable quad equations
• Lagrangian multiforms on Lie groups and non-commuting flows
• Hamiltonian structures for integrable hierarchies of Lagrangian PDEs
• Semi-discrete Lagrangian 2-forms and the Toda hierarchy
• A Variational Principle for Discrete Integrable Systems
• Algebraic entropy for hex systems
• Lagrangian multiforms and dispersionless integrable systems
• On the classification of discrete Hirota-type equations in 3D
• Solitons of a simple nonlinear model on the cubic lattice