On a singular perturbation in the linear soliton theory

Abstract

Initial-boundary value problems for the linearized Korteweg-de Vries equation with a small dispersion term are considered. Singularities of the corresponding families of the Green and Poisson operators are analysed as the small parameter vanishes. Asymptotic formulae for the eigenvalues and eigenfunctions of the corresponding singularly perturbed operator in the space variable are derived and justified. These formulae yield the conclusion that the dispersive nature of the linearized Korteweg-de Vries equation (in the case of the Cauchy problem) is preserved asymptotically for some perturbation terms in the case of mixed initial-boundary value problems too, when the small parameter goes to zero taking some explicitly specified discrete values.