A unified approach to Darboux transformations

Details A unified approach to Darboux transformations A unified approach to Darboux transformations

2009-05-05

arxiv.org/abs/0905.0520v2

Inverse Problems 25 (2009) 105003

Nonlinear Sciences

Exactly Solvable and Integrable Systems

final version that will appear in Inverse Problems

Scientific paper

10.1088/0266-5611/25/10/105003

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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