Integrable couplings of a generalized D-Kaup-Newell hierarchy and their Hamiltonian structures

TL;DR

This paper develops new integrable couplings of a generalized D-Kaup-Newell hierarchy, computes their Hamiltonian structures, and demonstrates their Liouville integrability with infinitely many symmetries and conserved quantities.

Contribution

It introduces enlarged spectral problems for the generalized D-KN hierarchy, constructs integrable couplings, and derives their Hamiltonian and bi-Hamiltonian structures.

Findings

Constructed integrable couplings via enlarged spectral problems.

Derived Hamiltonian and bi-Hamiltonian structures for the couplings.

Proved the hierarchies are Liouville integrable with infinite symmetries.

Abstract

We enlarge the spectral problem of a generalized D-Kaup-Newell (D-KN) spectral problem. Solving the enlarged zero-curvature equations, we produce integrable couplings. A reduction of the spectral matrix leads to a second integrable coupling system. Next, bilinear forms that are symmetric, ad-invariant, and non-degenerate on the given non-semisimple matrix Lie algebra are computed to employ the variational identity. The variational identity is then applied to the original enlarged spectral problem of a generalized D-KN hierarchy and the reduced problem. Hamiltonian structures are presented, as well as a bi-Hamiltonian formulation of the reduced problem. Both hierarchies have infinitely many commuting symmetries and conserved densities, i.e., are Liouville integrable.