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Generalized D-Kaup-Newell Integrable Systems And Their Integrable Couplings And Darboux Transformations, Morgan Ashley Mcanally

USF Tampa Graduate Theses and Dissertations

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 …