What is integrability of discrete variational systems?

Area of Science:

• Mathematical Physics
• Integrable Systems
• Variational Calculus

Background:

• Extends research on discrete integrable Lagrangian systems.
• Connects to pluriharmonic functions, Z-invariant models, and variational symmetries.
• Builds upon Lobb and Nijhoff's 2009 work on discrete integrable systems.

Purpose of the Study:

• Introduce and define the concept of a pluri-Lagrangian problem.
• Derive and analyze the corner equations for discrete pluri-Lagrangian systems.
• Investigate the consistency of these systems and their relation to integrable quad-equations.

Main Methods:

• Formulation of d-dimensional pluri-Lagrangian problems.
• Derivation of multi-time Euler-Lagrange equations (corner equations) for d=2.
• Analysis of corner equations for specific classes of two-forms, including those related to the ABS list.

Main Results:

• Derived the corner equations for discrete pluri-Lagrangian problems with d=2.
• Established the consistency of corner equations for specific two-forms.
• Demonstrated that two-forms are closed on solutions of corner equations, not just quad-equations.
• Identified a pluri-Lagrangian system not originating from a multi-dimensionally consistent quad-equation system.

Conclusions:

• Pluri-Lagrangian problems offer a framework for multi-dimensional consistency in variational systems.
• The corner equations are a key component in understanding discrete pluri-Lagrangian systems.
• The study bridges conceptual gaps in existing research and reveals new connections within integrable systems.