We show that well-chosen Lagrangians for a class of two-dimensional integrable lattice equations obey a closure relation when embedded in a higher dimensional lattice. On the basis of this property we formulate a Lagrangian description for such systems in terms of Lagrangian multiforms. We discuss the connection of this formalism with the notion of multidimensional consistency, and the role of the lattice from the point of view of the relevant variational principle.
We study the Lagrange formalism of the (rational) Ruijsenaars-Schneider (RS) system, both in discrete time as well as in continuous time, as a further example of a Lagrange 1-form structure in the sense of the recent paper [28]. The discrete-time model of the RS system was established some time ago arising via an Ansatz of a Lax pair, and was shown to lead to an exactly integrable correspondence (multivalued map) [17]. In this paper we consider an extended system representing a family of commuting flows of this type, and establish a connection with the lattice KP system. In the Lagrangian 1-form structure of this extended model, the closure relation is verified making use of the equations of motion. Performing successive continuum limits on the RS system, we establish the Lagrange 1-form structure for the corresponding continuum case of the RS model.
We present a Lagrangian for the bilinear discrete KP (or Hirota-Miwa) equation. Furthermore, we show that this Lagrangian can be extended to a Lagrangian 3-form when embedded in a higher dimensional lattice, obeying a closure relation. Thus we establish the multiform structure as proposed in [8] in a higher dimensional case.
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.
We present a hierarchy of discrete systems whose first members are the lattice modified Korteweg-de Vries equation, and the lattice modified Boussinesq equation. The N -th member in the hierarchy is an N -component system defined on an elementary plaquette in the 2-dimensional lattice. The system is multidimensionally consistent and a Lagrangian which respects this feature, i.e., which has the desirable closure property, is obtained.
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