Multiform description of the AKNS hierarchy and classical r-matrix

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 unravelled. 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.

Automated summary

Recent studies have discovered new properties of space-time duality in certain integrable classical field theories, leading to their reformulation using ideas from covariant Hamiltonian field theory. By extending these results to the whole hierarchy, specifically focusing on the AKNS hierarchy, and introducing a Lagrangian multiform, important objects such as a symplectic multiform and Hamiltonian multiform were explicitly constructed. These constructions help prove crucial results, including the rational classical r-matrix structure, multiform Hamilton equations, and a method to characterize an infinite set of conservation laws akin to the familiar criterion for a first integral.