Constraints and Soliton Solutions for KdV Hierarchy and AKNS Hierarchy

It is well-known that the finite-gap solutions of the KdV equationcan be generated by its recursion operator.We generalize the result to a special form of Lax pair,from which a method to constrain the integrable system to alower-dimensional or fewer variable integrable system is proposed.A direct result is that the n-soliton solutions of the KdV hierarchy can be completely depictedby a series of ordinary differential equations (ODEs), which may be gotten by a simple but unfamiliar Lax pair. Furthermore the AKNS hierarchy is constrained to a series of univariate integrable hierarchies. The key is a special form of Lax pair for the AKNS hierarchy. It is proved that under the constraints all equations of the AKNS hierarchy are linearizable.