Tuncay Aktosun and Cornelis van der Mee 2009 Inverse Problems 25 105003 doi:10.1088/0266-5611/25/10/105003
Tuncay Aktosun1 and Cornelis van der Mee2
Show affiliationsWe 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.
47E05 Ordinary differential operators (See also 34Bxx, 34Lxx)
35Q55 NLS-like (nonlinear Schrödinger) equations (See also 37K10)
Issue 10 (October 2009)
Received 9 April 2009, in final form 3 July 2009
Published 16 September 2009
Tuncay Aktosun and Cornelis van der Mee 2009 Inverse Problems 25 105003
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