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

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 oper ators can be factored into the integro-differential operator which generate s this hierarchy and the linearization operator of the KdV equation. An imp ortant consequence of this operator structure is that the linearization ope rators of all KdV hierarchy equations expanded around single-soliton soluti ons share the same complete set of eigenfunctions. Furthermore, these eigen functions are simply related to squared eigenstates of the Schrodinger oper ator with a soliton potential. Similar results hold for the adjoint lineari zation operators of this hierarchy. (C) 2001 Elsevier Science B.V. All righ ts reserved.