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.
- Publication:
-
Journal of Mathematical Physics
- Pub Date:
- December 2020
- DOI:
- 10.1063/5.0012153
- arXiv:
- arXiv:2004.01164
- Bibcode:
- 2020JMP....61l3506C
- Keywords:
-
- Mathematical Physics;
- High Energy Physics - Theory;
- Nonlinear Sciences - Exactly Solvable and Integrable Systems
- E-Print:
- v3: 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