Complete eigenfunctions of linearized integrable equations expanded around a soliton solution (Q2738207)

To develop a direct perturbation theory around a soliton, one needs a full set of solutions to the underlying equation linearized about the exact soliton solution. It was known that, provided that the underlying solution is amenable to the application of the inverse scattering transform, a full set of the eigenmodes of the linearized equation is related to squares of solutions to the associated scattering problem (which is based on the Zakharov-Shabat equations in the cases considered in this paper). A full set of the eigenmodes of the linearized equation for one-soliton solutions to hierarchies of integrable equations of the Korteweg - de Vries (KdV), nonlinear Schrödinger, and modified-KdV types is constructed in this paper. Inner products of the eigenmodes, which is a necessary technical ingredient of the perturbation theory to be based on the full set of the eigenmodes, are calculated too. It is shown that the around-soliton linearization operators generated by all the integrable equations belonging to one hierarchy share a common full set of the eigenmodes, and they are indeed expressed in terms of squared solutions to the Zakharov-Shabat equations, corresponding to the one-soliton state.