Hamiltonian multiform description of an integrable hierarchy
Abstract
Motivated by the notion of Lagrangian multiforms, which provide a Lagrangian formulation of integrability, and by the 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 + 1dimensional 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 the starting point, we provide a systematic construction of a Hamiltonian multiform based on a generalization of techniques of covariant Hamiltonian field theory. This also produces two other important objects: a symplectic multiform and the related multitime 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 Kortewegde Vries hierarchy, the sineGordon hierarchy (in lightcone coordinates), and the AblowitzKaupNewellSegur 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
 EPrint:
 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