A unified approach to Darboux transformations
Exactly Solvable and Integrable Systems
2009-09-18 v2
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 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 wave function change when a discrete eigenvalue is added to the spectrum.
Cite
@article{arxiv.0905.0520,
title = {A unified approach to Darboux transformations},
author = {Tuncay Aktosun and Cornelis van der Mee},
journal= {arXiv preprint arXiv:0905.0520},
year = {2009}
}
Comments
final version that will appear in Inverse Problems