Completeness of squared eigenfunctions of the Zakharov-Shabat spectral problem

Abstract

The completeness of eigenfunctions for linearized equations is critical for many applications, such as the study of stability of solitary waves. In this thesis, we work with the Nonlinear Schr{\"o}dinger (NLS) equation, associated with the Zakharov-Shabat spectral problem. Firstly, we construct a complete set of eigenfunctions for the spectral problem. Our method involves working with an adjoint spectral problem and deriving completeness and orthogonality relations between eigenfunctions and adjoint eigenfunctions. Furthermore, we prove completeness of squared eigenfunctions, which are used to represent solutions of the linearized NLS equation. For this, we find relations between the variation of potential and the variation of scattering data. Moreover, we show the connection between the squared eigenfunctions of the Zakharov-Shabat spectral problem and solutions of the linearized NLS equation.