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arXiv · 0905.0520

A unified approach to Darboux transformations

Tuncay Aktosun·Cornelis van der Mee
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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.

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Tuncay Aktosun, Cornelis van der Mee. 2009-07-21. A unified approach to Darboux transformations. https://doi.org/10.1088/0266-5611%2F25%2F10%2F105003

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