Constraints For KdV Hierarchy And AKNS Hierarchy

It is well-known that the finite-gap solutions of the KdV equation can 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 a lower-dimensional or fewer variable integrable system is proposed. A direct result is that the n-soliton solutions of the KdV hierarchy can be completely depicted by 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.The thesis is arranged as follows:Chapter 1 Introduction. Several methods for obtaining exact solutions or constraints for nonlinear mathematical physics equations are introduced in this chapter.Chapter 2 The method of IMs.Chapter 3 Soliton solutions of KdV hierarchy.Chapter 4 The nonlinearization of AKNS hierarchy.