Decomposition Of A Hierarchy Discrete Integrable Models And Straightening Out Of The Flows

A new discrete spectral problem is proposed, and nonlinear differential-difference equations of the corresponding hierarchy are obtained. It is interesting that the continuous limits of the second nontrivial differential-difference equations in the hierarchy are DNLS equations. The integrable symplectic map with its conserved integrals is obtained through the nonlinearization approach of eigenvalue problems. A generating function approach is introduced to prove the involutivity of conserved integrals and its functional independence. It is shown that the sympletctic map is completely integrable in the Liouville sense. Nonlinear differential-difference equations of the hierarchy are decomposed into Hamiltonian differential equations and a integrable symplectic map. Then the Abel-Jacobi coordinates are introduced to straighten out the corresponding flows, including continuous flow and the discrete flow.