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Eigenfunctions of Linearized Integrable Equations Expanded Around an Arbitrary Solution

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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 integrodifferential operator that 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/multisoliton perturbation theory and the inverse-scattering based perturbation theory for these hierarchy equations.
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Document Type: Original Article

Affiliations: University of Vermont

Publication date: January 1, 2002

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