Lagrangian multiform structure for the lattice Gel'fand-Dikii hierarchy

TL;DR

This paper develops a Lagrangian multiform structure for the lattice Gel'fand-Dikii hierarchy, extending the geometric framework to multi-component systems and revealing new integrability properties.

Contribution

It introduces a Lagrangian 2-form for the hierarchy and demonstrates its closure relation in a higher-dimensional lattice, expanding the multiform structure to multi-component systems.

Findings

Lagrangian for the lattice Gel'fand-Dikii hierarchy is constructed.

The multiform structure obeys a closure relation in higher dimensions.

Extension of the multiform framework to multi-component systems.

Abstract

The lattice Gel'fand-Dikii hierarchy was introduced by Nijhoff, Papageorgiou, Capel and Quispel in 1992 as the family of partial difference equations generalizing to higher rank the lattice Korteweg-de Vries systems, and includes in particular the lattice Boussinesq system. We present a Lagrangian for the generic member of the lattice Gel'fand-Dikii hierarchy, and show that it can be considered as a Lagrangian 2-form when embedded in a higher dimensional lattice, obeying a closure relation. Thus the multiform structure proposed in arXiv:0903.4086v2 [nlin.SI] is extended to a multi-component system.