On the Lagrangian structure of integrable hierarchies

Abstract

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 systems. We derive the multi-time Euler Lagrange equations in their full generality for hierarchies of two-dimensional systems, and construct a pluri-Lagrangian formulation of the potential Korteweg-de Vries hierarchy.

29 pages

References in corpus (2)

• Faddeev-Volkov solution of the Yang-Baxter Equation and Discrete Conformal Symmetry
• Discrete pluriharmonic functions as solutions of linear pluri-Lagrangian systems

Cited by in corpus (18)

• A Variational Approach to Lax Representations
• Variational symmetries and pluri-Lagrangian systems in classical mechanics
• Variational symmetries and Lagrangian multiforms
• Lagrangian multiforms for Kadomtsev-Petviashvili (KP) and the Gelfand-Dickey hierarchy
• Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs
• Continuum limits of pluri-Lagrangian systems
• Lagrangian 3-form structure for the Darboux system and the KP hierarchy
• Classical Yang-Baxter equation, Lagrangian multiforms and ultralocal integrable hierarchies
• Hamiltonian multiform description of an integrable hierarchy
• A variational perspective on continuum limits of ABS and lattice GD equations
• Lagrangian multiforms on coadjoint orbits for finite-dimensional integrable systems
• Hamiltonian formulation and aspects of integrability of generalised hydrodynamics
• Hamiltonian structures for integrable hierarchies of Lagrangian PDEs
• A Variational Principle for Discrete Integrable Systems
• Commutativity in Lagrangian and Hamiltonian Mechanics
• On the geometry of Lagrangian one-forms
• Lagrangian multiforms and dispersionless integrable systems
• Lagrangian Multiform for Cyclotomic Gaudin Models