A unified approach to Darboux transformations

Abstract

We analyze a certain class of integral equations related to Marchenko equations and Gel'fand–Levitan equations associated with various systems of ordinary differential operators. When the integral operator is perturbed by a finite-rank perturbation, we explicitly evaluate the change in the solution. We show how this result provides a unified approach to derive Darboux transformations associated with various systems of ordinary differential operators. We illustrate our theory by deriving the Darboux transformation for the Zakharov–Shabat system and show how the potential and wavefunction change when a discrete eigenvalue is added to the spectrum.