# Lagrangian multiforms and multidimensional consistency

@article{Lobb2009LagrangianMA, title={Lagrangian multiforms and multidimensional consistency}, author={Sarah Lobb and Frank W Nijhoff}, journal={Journal of Physics A}, year={2009}, volume={42}, pages={454013} }

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.

#### 59 Citations

On the Lagrangian Structure of Integrable Hierarchies

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- 2016

We develop the concept of pluri-Lagrangian structures for integrable hierarchies. This is a continuous counterpart of the pluri-Lagrangian (or Lagrangian multiform) theory of integrable lattice… Expand

Variational symmetries and Lagrangian multiforms

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- Letters in Mathematical Physics
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By considering the closure property of a Lagrangian multiform as a conservation law, we use Noether’s theorem to show that every variational symmetry of a Lagrangian leads to a Lagrangian multiform.… Expand

An integrable multicomponent quad-equation and its Lagrangian formulation

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- 2012

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 Nth member in the hierarchy is an… Expand

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- Journal of Geometry and Physics
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Abstract It is shown that the Zakharov–Mikhailov (ZM) Lagrangian structure for integrable nonlinear equations derived from a general class of Lax pairs possesses a Lagrangian multiform structure in… Expand

On Discrete Integrable Equations with Convex Variational Principles

- Mathematics, Physics
- 2012

The Lagrangian structure of two-dimensional integrable systems on quad-graphs is investigated. We give reality conditions under which the action functionals are strictly convex. In particular, this… Expand

FAST TRACK COMMUNICATION: Lagrangian multiform structure for the lattice KP system

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- 2009

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… Expand

On the Lagrangian Structure of Integrable Quad-Equations

- Mathematics, Physics
- 2010

The new idea of flip invariance of action functionals in multidimensional lattices was recently highlighted as a key feature of discrete integrable systems. Flip invariance was proved for several… Expand

On non-multiaffine consistent-around-the-cube lattice equations

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- 2012

Abstract We show that integrable involutive maps, due to the fact they admit three integrals in separated form, can give rise to equations, which are consistent around the cube and which are not in… Expand

On integrability of discrete variational systems. Octahedron relations

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- 2014

We elucidate consistency of the so-called corner equations which are elementary building blocks of Euler-Lagrange equations for two-dimensional pluri-Lagrangian problems. We show that their… Expand

A Variational Perspective on Continuum Limits of ABS and Lattice GD Equations

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- Symmetry, Integrability and Geometry: Methods and Applications
- 2019

A pluri-Lagrangian structure is an attribute of integrability for lattice equations and for hierarchies of differential equations. It combines the notion of multi-dimensional consistency (in the… Expand

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