A unified approach to Darboux transformations

TL;DR

This paper presents a unified framework for Darboux transformations across different differential operator systems by analyzing integral equations and their perturbations, with explicit formulas and applications to the Zakharov-Shabat system.

Contribution

It introduces a general approach to Darboux transformations using integral equation perturbation analysis, applicable to various differential systems.

Findings

Explicit formulas for potential and wave function changes when adding eigenvalues.

Unified method applicable to multiple differential operator systems.

Application to the Zakharov-Shabat system demonstrating practical use.

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.