Complete eigenfunctions of linearized integrable equations expanded around an arbitrary solution

Abstract

Complete eigenfunctions of linearized integrable equations expanded around an arbitrary solution are obtained for the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy and the Korteweg-de Vries (KdV) hierarchy. It is shown that the linearization operators and the recursion operator which generates the hierarchy are commutable. Consequently, eigenfunctions of the linearization operators are precisely squared eigenfunctions of the associated eigenvalue problem. Similar results are obtained for the adjoint linearization operators as well. These results make a simple connection between the direct soliton/multi-soliton perturbation theory and the inverse-scattering based perturbation theory for these hierarchy equations.