  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.
