Binary nonlinearization of spectral problems of the perturbation AKNS systems

The method of nonlinearization of spectral problems is extended to the perturbation AKNS systems, and a new kind of finite-dimensional Hamiltonian systems is obtained. It is shown that the obtained Hamiltonian systems are just the perturbation systems of the well-known constrained AKNS flows and thus their Liouville integrability is established by restoring from the Liouville integrability of the constrained AKNS flows. As a byproduct, the process of binary nonlinearization of spectral problems and the process of perturbation of soliton equations commute in the case of the AKNS hierarchy.