Hamiltonian multiform description of an integrable hierarchy
Vincent Caudrelier, Matteo Stoppato
备注：
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
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.
In recent years, new properties of space-time duality in the Hamiltonian formalism of certain integrable classical field theories have been discovered and have led to their reformulation using ideas from covariant Hamiltonian field theory: in this sense, the covariant nature of their classical $r$-matrix structure was unraveled. Here, we solve the open question of extending these results to a whole hierarchy. We choose the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy. To do so, we introduce for the first time a Lagrangian multiform for the entire AKNS hierarchy. We use it to construct explicitl…
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