Multiform description of the AKNS hierarchy and classical r-matrix

Abstract In recent years, new properties of space-time duality in the Hamiltonian formalism certain integrable classical field theories have been discovered and led to their reformulation using ideas from covariant theory: this sense, nature r -matrix structure was unravelled. Here, we solve open question extending these results a whole hierarchy. We choose Ablowitz–Kaup–Newell–Segur (AKNS) To do so, introduce for first time Lagrangian multiform entire AKNS use it construct explicitly necessary objects introduced previously by us: symplectic multiform, multi-time Poisson bracket multiform. Equipped with these, prove following results: (i) Lax form containing sequence matrices hierarchy possesses rational structure; (ii) zero curvature equations are Hamilton associated our bracket; (iii) provides way characterise infinite set conservation laws reminiscent familiar criterion { I , H } = 0 integral .