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

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.

12 pages

Cited by in corpus (20)

• Variational formulation of commuting Hamiltonian flows: multi-time Lagrangian 1-forms
• Liouville-Arnold integrability of the pentagram map on closed polygons
• What is integrability of discrete variational systems?
• An integrable multicomponent quad equation and its Lagrangian formulation
• Non-commutative lattice modified Gel'fand-Dikii systems
• Variational symmetries and pluri-Lagrangian systems in classical mechanics
• On the Lagrangian formulation of multidimensionally consistent systems
• Symmetries and conservation laws of lattice Boussinesq equations
• Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Relativistic Toda-type systems
• Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs
• Discrete pluriharmonic functions as solutions of linear pluri-Lagrangian systems
• Discrete time Toda systems
• Multiform description of the AKNS hierarchy and classical r-matrix
• A variational perspective on continuum limits of ABS and lattice GD equations
• Hamiltonian multiform description of an integrable hierarchy
• Lagrangian multiforms on Lie groups and non-commuting flows
• A Variational Principle for Discrete Integrable Systems
• On the Lagrangian 1-Form Structure of the Hyperbolic Calogero-Moser System
• On the Lagrangian multiform structure of the extended lattice Boussinesq system
• On the Fourth-Order Lattice Gel'fand-Dikii Equations