Abstract
Starting from a 3×3 matrix spectral problem, we derivea hierarchy of nonlinear equations. It is shown that the hierarchypossesses bi-Hamiltonian structure. Under the symmetry constraintsbetween the potentials and the eigenfunctions, Lax pair and adjointLax pairs including partial part and temporal part are nonlineariedinto two finite-dimensional Hamiltonian systems (FDHS) in Liouvillesense. Moreover, an explicit N-fold Darboux transformation for CDNSequation is constructed with the help of a gauge transformation ofthe spectral problem.
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