Integrable generalizations of the two new soliton hierarchies of AKNS and KN types associated with $so(3,\mathbb{R})$

TL;DR

This paper generalizes AKNS and KN soliton hierarchies linked to so(3,R), deriving new integrable systems with explicit recursion operators and bi-Hamiltonian structures, confirming their Liouville integrability.

Contribution

It introduces integrable generalizations of AKNS and KN hierarchies associated with so(3,R), including explicit recursion operators and bi-Hamiltonian structures.

Findings

Derived generalized soliton hierarchies using zero curvature formulation

Constructed explicit recursion operators and bi-Hamiltonian structures

Confirmed Liouville integrability of the new hierarchies

Abstract

The two matrix spectral problems of Ablowitz-Kaup-Newell-Segur (AKNS) and Kaup-Newell (KN) types associated with so(3,R) are generalized. The corresponding hierarchies of generalized soliton equations are derived by the standard procedure using the zero curvature formulation. Recursion operators and bi-Hamiltonian structures are explicitly constructed for the resulting two generalized soliton hierarchies of AKNS and KN types, which shows their Liouville integrability.