Multiform description of the AKNS hierarchy and classical r-matrix

Abstract

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 explicitly the necessary objects introduced previously by us: a symplectic multiform, a multi-time Poisson bracket and a Hamiltonian multiform. Equipped with these, we prove the following results: $(i)$ the Lax form containing the whole sequence of Lax matrices of the hierarchy possesses the rational classical $r$-matrix structure; $(ii)$ The zero curvature equations of the AKNS hierarchy are multiform Hamilton equations associated to our Hamiltonian multiform and multi-time Poisson bracket; $(iii)$ The Hamiltonian multiform provides a way to characterise the infinite set of conservation laws of the hierarchy reminiscent of the familiar criterion $\{I,H\}=0$ for a first integral $I$.

arXiv comment34 pages; accepted version

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• Classical Yang-Baxter equation, Lagrangian multiforms and ultralocal integrable hierarchies
• Hamiltonian multiform description of an integrable hierarchy
• Lagrangian multiforms on coadjoint orbits for finite-dimensional integrable systems
• Lagrangian multiforms on Lie groups and non-commuting flows
• Hamiltonian formulation and aspects of integrability of generalised hydrodynamics
• Quantum integrability: Lagrangian 1-form case
• On the geometry of Lagrangian one-forms