Bi-integrable couplings of a Kaup-Newell type soliton hierarchy and their bi-Hamiltonian structures

A new Kaup-Newell type soliton hierarchy is generated from an asymmetric matrix spectral problem associated with the three-dimensional special linear Lie algebra sl(2,R). Then based on semi-direct sums of matrix Lie algebras consisting of 3×3 block matrix Lie algebras, corresponding bi-integrable couplings of this hierarchy are constructed. Each equation in the resulting system has a bi-Hamiltonian structure furnished by the variational identity, which lead to Liouville integrability.