2010/08/11 by Pavlos Xenitidis, Frank Nijhoff, Sarah Lobb · 1 citation
Physics and Astronomy · Mathematics · #nlin.SI #math-ph #math.MP #math.SG
paper · pdf · doi:10.1098/rspa.2011.0124
22 pages
arxiv created 2010/08/11 · arxiv updated 2015/05/19
Multidimensional consistency has emerged as a key integrability property for partial difference equations (PΔEs) defined on the "space-time" lattice. It has led, among other major insights, to a classification of scalar affine-linear quadrilateral PΔEs 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 PΔE 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 PΔEs 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.