On the Lagrangian formulation of multidimensionally consistent systems

Abstract

Multidimensional consistency has emerged as a key integrability property for partial difference equations (PDEs) defined on the 'space-time' lattice. It has led, among other major insights, to a classification of scalar affine-linear quadrilateral PDEs possessing this property, leading to the so-called Adler-Bobenko-Suris (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 non-trivial property of Lagrangians for those integrable lattice equations, namely the fact that on the solutions of the corresponding PDE the Lagrange forms are closed, i.e. they obey a closure relation. Here, we extend those results to the continuous case: it is known that associated with the integrable PDEs there exist systems of partial differential equations (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.

Quantum variational principle and quantum multiform structure: The case of quadratic Lagrangians

“…The Lagrangian itself is a critical point of the classical variational principle over surfaces: it obeys the closure property on the classical equations of motion, such that the surface can be allowed to freely vary under local moves. Indeed, it is also fairly unique, as seen in (8).…”

Continuum limits of pluri-Lagrangian systems

“…A pluri-Lagrangian description of Equation (29) was found in [13], the Lagrange function itself goes back to [8]. It reads…”

“…By recovering a pluri-Lagrangian structure on the continuous side, 1 The author prefers the term "pluri-Lagrangian" over "Lagrangian multiform" because it is not the differential form that has a multiplicity or plurality to it, but rather its interpretation as a Lagrangian. There is a minor distinction in how both names have been used in the literature: "pluri-Lagrangian" indicates that solutions are critical with respect to variations of the dependent variable on any fixed surface [5,7,24], whereas "Lagrangian multiform" is mostly used when one also requires criticality with respect to variations in the geometry of the surface [12,13,29,30]. This distinction is not relevant to the present work.…”

“…z n,m (t, s) = v x 1Because of this it has become known as the generating PDE[13,29] for the SKdV hierarchy. Renaming the parameters t = λ 2 1 and s = λ 2 2 we obtain once again the odd order Miwa shifts z n,m = v x 1…”

A variational approach to Lax representations

“…d = 2 for systems of 1+1-dimensional equations). This led to the introduction of a new notion of a Lagrangian multiform [1,3], where the multidimensional consistency manifests itself by the Euler-Lagrange (EL) equations of the Lagrangian d-form being independent of the choice of the surface of integration in the action functional.…”