On the Lagrangian formulation of multidimensionally consistent systems

TL;DR

This paper extends the concept of multidimensional consistency and Lagrangian multi-form structures from discrete lattice equations to continuous PDE systems, establishing a universal Lagrangian framework with a closure property.

Contribution

It introduces a universal Lagrangian structure for affine-linear quad-lattices and their continuous PDE counterparts, demonstrating the closure property in both cases.

Findings

Universal Lagrange structure for affine-linear quad-lattices

Lagrangian multi-form structure for continuous PDEs

Closure property of Lagrangian forms in both discrete and continuous cases

Abstract

Multidimensional consistency has emerged as a key integrability property for partial difference equations (P$\Delta$Es) defined on the "space-time" lattice. It has led, among other major insights, to a classification of scalar affine-linear quadrilateral P$\Delta$Es possessing this property, leading to the so-called ABS list. Recently, a new variational principle has been proposed that describes the multidimensional consistency in terms of discrete Lagrangian multi-forms. This description is based on a fundamental and highly nontrivial property of Lagrangians for those integrable lattice equations, namely the fact that on the solutions of the corresponding P$\Delta$E the Lagrange forms are closed, i.e. they obey a {\it closure relation}. Here we extend those results to the continuous case: it is known that associated with the integrable P$\Delta$Es there exist systems of PDEs, in fact differential equations with regard to the parameters of the lattice as independent variables, which equally possess the property of multidimensional consistency. In this paper we establish a universal Lagrange structure for affine-linear quad-lattices alongside a universal Lagrange multi-form structure for the corresponding continuous PDEs, and we show that the Lagrange forms possess the closure property.