Complete eigenfunctions of linearized integrable equations expanded around an arbitrary solution

摘要

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