Generalized D-Kaup-Newell integrable systems and their integrable couplings and Darboux transformations

We present a new spectral problem, a generalization of the D-Kaup-Newell spectral problem, associated with the Lie algebra sl(2,R). Zero curvature equations furnish the soliton hierarchy. The trace identity produces the Hamiltonian structure for the hierarchy. Lastly, a reduction of the spectral problem is shown to have a different soliton hierarchy with a bi-Hamiltonian structure. The first major motivation of this dissertation is to present spectral problems that generate two soliton hierarchies with infinitely many commuting conservation laws and high-order symmetries, i.e., they are Liouville integrable. We use the soliton hierarchies and a non-seimisimple matrix loop Lie algebra in order to construct integrable couplings. An enlarged spectral problem is presented starting from a generalization of the D-Kaup-Newell spectral problem. Then the enlarged zero curvature equations are solved from a series of Lax pairs producing the desired integrable couplings. A reduction is made of the original enlarged spectral problem generating a second integrable coupling system. Next, we discuss how to compute bilinear forms that are symmetric, ad-invariant, and non-degenerate on the given non-semisimple matrix Lie algebra to employ the variational identity. The variational identity is applied to the original integrable couplings of a generalized D-Kaup-Newell soliton hierarchy to furnish its Hamiltonian structures. Then we apply the variational identity to the reduced integrable couplings. The reduced coupling system has a bi-Hamiltonian structure. Both integrable coupling systems retain the properties of infinitely many commuting high-order symmetries and conserved densities of their original subsystems and, again, are Liouville integrable. In order to find solutions to a generalized D-Kaup-Newell integrable coupling system, a theory of Darboux transformations on integrable couplings is formulated. The theory pertains to a spectral problem where the spectral matrix is a polynomial in lambda of any order. An application to a generalized D-Kaup-Newell integrable couplings system is worked out, along with an explicit formula for the associated Bäcklund transformation. Precise one-soliton-like solutions are given for the m-th order generalized D-Kaup-Newell integrable coupling system