• DOI:10.1063/5.0012153
• Corpus ID: 214774809

Hamiltonian multiform description of an integrable hierarchy

@article{Caudrelier2020HamiltonianMD,
  title={Hamiltonian multiform description of an integrable hierarchy},
  author={Vincent Caudrelier and Matteo Stoppato},
  journal={arXiv: Mathematical Physics},
  year={2020}
}

• V. Caudrelier, Matteo Stoppato
• Published 2 April 2020
• Mathematics
• arXiv: Mathematical Physics

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.