Hamiltonian Systems and Darboux Transformation Associated with a 3×3 Matrix Spectral Problem

Starting from a 3×3 matrix spectral problem,we derive a hierarchy of nonlinear equations.It is shownthat the hierarchy possesses bi-Hamiltonian structure.Under the symmetry constraints between the potentials and theeigenfunctions,Lax pair and adjoint Lax pairs including partial part and temporal part are nonlinearied into two finite-dimensional Hamiltonian systems(FDHS)in Liouville sense.Moreover,an explicit N-fold Darboux transformation forCDNS equation is constructed with the help of a gauge transformation of the spectral problem.