A variational approach to Lax representations

  title={A variational approach to Lax representations},
  author={Duncan Sleigh and Frank W. Nijhoff and Vincent Caudrelier},
  journal={Journal of Geometry and Physics},

Variational symmetries and Lagrangian multiforms

  • Duncan SleighFrank NijhoffVincent Caudrelier
  • Mathematics
    Letters in Mathematical Physics
  • 2019
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.

Multiform description of the AKNS hierarchy and classical r-matrix

In recent years, new properties of space-time duality in the Hamiltonian formalism of certain integrable classical field theories have been discovered and have led to their reformulation using ideas

Hamiltonian structures for integrable hierarchies of Lagrangian PDEs

  • Mats Vermeeren
  • Mathematics
    Open Communications in Nonlinear Mathematical Physics
  • 2021
Many integrable hierarchies of differential equations allow a variational description, called a Lagrangian multiform or a pluri-Lagrangian structure. The fundamental object in this theory is not a

Lagrangian 3-form structure for the Darboux system and the KP hierarchy

  • F. Nijhoff
  • Computer Science
    Letters in Mathematical Physics
  • 2023
A Lagrangian multiform structure is established for a generalisation of the Darboux system describing orthogonal curvilinear coordinate systems for the continuous Kadomtsev–Petviashvili (KP) hierarchy.

Semi-discrete Lagrangian 2-forms and the Toda hierarchy

We present a variational theory of integrable differential-difference equations (semi-discrete integrable systems). This is an extension of the ideas known by the names ‘Lagrangian multiforms’ and

Lagrangian Multiforms for Kadomtsev–Petviashvili (KP) and the Gelfand–Dickey Hierarchy

We present, for the first time, a Lagrangian multiform for the complete Kadomtsev–Petviashvili hierarchy—a single variational object that generates the whole hierarchy and encapsulates its

Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs

We investigate the relation between pluri-Lagrangian hierarchies of 2-dimensional partial differential equations and their variational symmetries. The aim is to generalize to the case of partial

Classical Yang-Baxter equation, Lagrangian multiforms and ultralocal integrable hierarchies

We cast the classical Yang-Baxter equation (CYBE) in a variational context for the first time, by relating it to the theory of Lagrangian multiforms, a framework designed to capture integrability in



Lagrangian multiforms and multidimensional consistency

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

On the Lagrangian formulation of multidimensionally consistent systems

  • P. XenitidisF. NijhoffS. Lobb
  • Mathematics, Computer Science
    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2011
A universal Lagrange structure for affine-linear quad-lattices alongside a universal Lagrangian multi-form structure for the corresponding continuous PDEs is established, and it is shown that the Lagrange forms possess the closure property.

On the Lagrangian Structure of Integrable Hierarchies

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

Symplectic forms in the theory of solitons

We develop a Hamiltonian theory for 2D soliton equations. In particular, we identify the spaces of doubly periodic operators on which a full hierarchy of commuting flows can be introduced, and show

On the integrable geometry of soliton equations and $N=2$ supersymmetric gauge theories

We provide a unified construction of the symplectic forms which arise in the solution of both N=2 supersymmetric Yang-Mills theories and soliton equations. Their phase spaces are Jacobian-type

Soliton Equations and Hamiltonian Systems

Integrable Systems Generated by Linear Differential nth Order Operators Hamiltonian Structures Hamiltonian Structure of the GD Hierarchies Modified KdV and GD. The Kupershmidt-Wilson Theorem The KP

General Zakharov-Shabat equations, multi-time Hamiltonian formalism, and constants of motion

We construct a Hamiltonian formalism for general Zakharov-Shabat equations (zero curvature equations with rational dependence on a parameter) as well as their constants of motion, and prove that the