Hamiltonian multiform description of an integrable hierarchy

Abstract

Motivated by the notion of Lagrangian multiforms, which provide a Lagrangian formulation of integrability, and by results of the authors on the role of covariant Hamiltonian formalism for integrable field theories, we propose the notion of Hamiltonian multiforms for integrable $1+1$-dimensional field theories. They provide the Hamiltonian counterpart of Lagrangian multiforms and encapsulate in a single object an arbitrary number of flows within an integrable hierarchy. For a given hierarchy, taking a Lagrangian multiform as starting point, we provide a systematic construction of a Hamiltonian multiform based on a generalisation of techniques of covariant Hamiltonian field theory. This also produces two other important objects: a symplectic multiform and the related multi-time Poisson bracket. They reduce to a multisymplectic form and the related covariant Poisson bracket if we restrict our attention to a single flow in the hierarchy. Our framework offers an alternative approach to define and derive conservation laws for a hierarchy. We illustrate our results on three examples: the potential Korteweg-de Vries hierarchy, the sine-Gordon hierarchy (in light cone coordinates) and the Ablowitz-Kaup-Newell-Segur hierarchy.

arXiv commentv3: Improved discussion after Proposition 2.2, with changes until the end of Section 2.3. Lemma 2.4 was added. References updated and typos corrected. Accepted for publication in J. Math. Phys

References in corpus (24)

• Lax pair for the Adler (lattice Krichever-Novikov) System
• Canonical Structure of Classical Field Theory in the Polymomentum Phase Space
• Integrable systems on quad-graphs
• Lagrangian multiforms and multidimensional consistency
• Variational formulation of commuting Hamiltonian flows: multi-time Lagrangian 1-forms
• Lagrangian and Hamiltonian structures in an integrable hierarchy and space-time duality
• Discrete-time Calogero-Moser system and Lagrangian 1-form structure
• Lagrangian multiform structure for the lattice Gel'fand-Dikii hierarchy
• A multisymplectic approach to defects in integrable classical field theory
• What is integrability of discrete variational systems?
• On the Lagrangian structure of integrable quad-equations
• A Variational Approach to Lax Representations
• On the Lagrangian structure of integrable hierarchies
• Variational symmetries and pluri-Lagrangian systems in classical mechanics
• Variational symmetries and Lagrangian multiforms
• On the Lagrangian formulation of multidimensionally consistent systems
• Multisymplectic approach to integrable defects in the sine-Gordon model
• 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
• Continuum limits of pluri-Lagrangian systems
• On the origin of dual Lax pairs and their $r$-matrix structure
• A connection between the classical r-matrix formalism and covariant Hamiltonian field theory
• Multiform description of the AKNS hierarchy and classical r-matrix
• Hamiltonian structures for integrable hierarchies of Lagrangian PDEs

Cited by in corpus (7)

• On the Zakharov-Mikhailov action: $4$d Chern-Simons origin and covariant Poisson algebra of the Lax connection
• Classical Yang-Baxter equation, Lagrangian multiforms and ultralocal integrable hierarchies
• Multiform description of the AKNS hierarchy and classical r-matrix
• Lagrangian multiforms on coadjoint orbits for finite-dimensional integrable systems
• Lagrangian multiforms on Lie groups and non-commuting flows
• Hamiltonian structures for integrable hierarchies of Lagrangian PDEs
• On the geometry of Lagrangian one-forms