Abstract

Complete eigenfunctions of linearized integrable equations
expanded around their solutions

Jianke Yang
Department of Mathematics and Statistics
University of Vermont

Complete eigenfunctions for an integrable equation linearized around its solution are the key to the development of a direct soliton or multi-soliton perturbation theory. In this talk, we construct such complete eigenfunctions for the Ablowitz-Kaup-Newell-Segur hierarchy and the Korteweg-de Vries hierarchy. By showing that this linearization operator and the recursion operator of the underlying hierarchy are commutable, we establish that the complete eigenfunctions of the linearization operator are precisely the squared eigenstates of the Zakharov-Shabat or Schroedinger system. For linearization around a soliton solution, our results go even further. We will show that the linearization operator for any member of a hierarchy can be factored into the recursion operator of the hierarchy and the linearization operator of the lowest-order equation in the hierarchy. Our results reveal that inverse-scattering-based soliton/multi-soliton perturbation theory and direct soliton/multi-soliton perturbation theory are equivalent.

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