Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Relativistic Toda-type systems

Abstract

We establish the pluri-Lagrangian structure for families of Bäcklund transformations of relativistic Toda-type systems. The key idea is a novel embedding of these discrete-time (one-dimensional) systems into certain two-dimensional pluri-Lagrangian lattice systems. This embedding allows us to identify the corner equations (which are the main building blocks of the multi-time Euler-Lagrange equations) with local superposition formulae for Bäcklund transformations. These superposition formulae, in turn, are key ingredients necessary to understand and to prove commutativity of the multi-valued Bäcklund transformations. Furthermore, we discover a two-dimensional generalization of the spectrality property known for families of Bäcklund transformations. This result produces a family of local conservations laws for two-dimensional pluri-Lagrangian lattice systems, with densities being derivatives of the discrete 2-form with respect to the Bäcklund (spectral) parameter. Thus, a relation of the pluri-Lagrangian structure with more traditional integrability notions is established.

arXiv comment25 pages

References in corpus (14)

• Classification of integrable equations on quad-graphs. The consistency approach
• Lax pair for the Adler (lattice Krichever-Novikov) System
• Few remarks on Baecklund transformations for many-body systems
• Integrable systems on quad-graphs
• Variational formulation of commuting Hamiltonian flows: multi-time Lagrangian 1-forms
• Discrete-time Calogero-Moser system and Lagrangian 1-form structure
• Lagrangian multiform structure for the lattice Gel'fand-Dikii hierarchy
• What is integrability of discrete variational systems?
• An integrable multicomponent quad equation and its Lagrangian formulation
• On the Lagrangian structure of integrable quad-equations
• Legendre transformations on the triangular lattice
• Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Toda-type systems
• Discrete pluriharmonic functions as solutions of linear pluri-Lagrangian systems
• On the Lagrangian structure of 3D consistent systems of asymmetric quad-equations

Cited by in corpus (8)

• Variational symmetries and pluri-Lagrangian systems in classical mechanics
• Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs
• Discrete time Toda systems
• Multiform description of the AKNS hierarchy and classical r-matrix
• Hamiltonian multiform description of an integrable hierarchy
• A Variational Principle for Discrete Integrable Systems
• Integrable Hamiltonian Hierarchies and Lagrangian 1-Forms
• Quantum integrability: Lagrangian 1-form case