Lagrangian structures and multidimensional consistency

Abstract

The conventional point of view is that the Lagrangian is a scalar object (or equivalently a volume form), which through the Euler-Lagrange equations provides us with one single equation (i.e., one per component of the dependent variable). Multidimensional consistency is a key integrability property of certain discrete systems; it implies that we are dealing with infinite hierarchies of compatible equations rather than with one single equation. Requiring the property of multidimensional consistency to be reflected also in the Lagrangian formulation of such systems, we arrive naturally at the construction of Lagrangian multiforms, i.e., Lagrangians which are the components of a form and satisfy a closure relation. We demonstrate that the Lagrangians of many important examples fit into this framework: the so-called ABS list of systems on quad graphs, which includes the discrete Korteweg-de Vries equation; the Gel'fand-Dikii hierarchy, which includes the discrete Boussinesq equation; and the bilinear discrete Kadomtsev-Petviashvili equation. On the basis of this we propose a new variational principle for integrable systems which brings in the geometry of the space of independent variables, and from this principle we can then derive any equation in the hierarchy. We also extend the notion of Lagrangian forms, and the corresponding new variational principle, to continuous systems, using the example of the generating partial dierential equation for the Korteweg-de Vries hierarchy.