Multiform description of the AKNS hierarchy and classical r-matrix

TL;DR

This paper extends the covariant Hamiltonian formalism and classical r-matrix structure to the entire AKNS hierarchy using a novel Lagrangian multiform, revealing new geometric and algebraic insights into integrable systems.

Contribution

It introduces the first Lagrangian multiform for the AKNS hierarchy and constructs associated multiform Hamiltonian structures, extending previous results to a whole integrable hierarchy.

Findings

The Lax matrices of the hierarchy have a rational classical r-matrix structure.

Zero curvature equations are shown to be multiform Hamilton equations.

The Hamiltonian multiform characterizes the hierarchy's infinite conservation laws.

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 $I$.