Complete eigenfunctions of linearized integrable equations expanded around a soliton solution

Abstract

Complete eigenfunctions for an integrable equation linearized around a soliton solution are the key to the development of a direct soliton perturbation theory. In this article, we explicitly construct such eigenfunctions for a large class of integrable equations including the KdV, NLS and mKdV hierarchies. We establish the striking result that the linearization operators of all equations in the same integrable hierarchy share the same complete set of eigenfunctions. Furthermore, these eigenfunctions are precisely the squared eigenfunctions of the associated eigenvalue problem. The key step in our derivation is to show that the linearization operator of an integrable equation can be factored into a function of the integro-differential operator which generates the integrable equation, and the linearization operator of the lowest-order integrable equation in the same hierarchy. We also obtain similar results for the adjoint linearization operator of an integrable equation. Even though our analysis is conducted only for the KdV, NLS and mKdV hierarchies, similar results are expected for other integrable hierarchies as well. We further explicitly present the complete eigenfunctions for the KdV, NLS and mKdV hierarchy equations and give their inner products, thus they can be readily used to develop a direct soliton perturbation theory for any of those hierarchy equations.