Structure of linearization operators of the Korteweg–de Vries hierarchy equations expanded around single-soliton solutions

Abstract

In this Letter, we analyze the structure of linearization operators of the Korteweg–de Vries (KdV) hierarchy equations expanded around single-soliton solutions. We uncover the remarkable property that these linearization operators can be factored into the integro-differential operator which generates this hierarchy and the linearization operator of the KdV equation. An important consequence of this operator structure is that the linearization operators of all KdV hierarchy equations expanded around single-soliton solutions share the same complete set of eigenfunctions. Furthermore, these eigenfunctions are simply related to squared eigenstates of the Schrödinger operator with a soliton potential. Similar results hold for the adjoint linearization operators of this hierarchy.