Darboux transformation for the NLS equation. (English) Zbl 1218.45005

Aktosun, Tuncay; van der Mee, Cornelis

Ma, Wen-Xiu (ed.) et al., Nonlinear and modern mathematical physics. Proceedings of the 1st international workshop held in Beijing, China, July 15–21, 2009. Melville, NY: American Institute of Physics (ISBN 978-0-7354-0755-8/pbk). AIP Conference Proceedings 1212, 254-263 (2010).

Summary: We analyze a certain class of integral equations associated with Marchenko equations and Gel’fand-Levitan equations. Such integral equations arise through a Fourier transformation on various ordinary differential equations involving a spectral parameter. When the integral operator is perturbed by a finite-rank perturbation, we explicitly evaluate the change in the solution in terms of the unperturbed quantities and the finite-rank perturbation. We show that this result provides a fundamental approach to derive Darboux transformations for various systems of ordinary differential operators.
We illustrate our theory by providing the explicit Darboux transformation for the Zakharov-Shabat system and show how the potential and wave function change when a simple discrete eigenvalue is added to the spectrum, and thus we also provide a one-parameter family of Darboux transformations for the nonlinear Schrödinger (NLS) equation.
For the entire collection see [Zbl 1205.00092].

MSC:

45E10	Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
35Q55	NLS equations (nonlinear Schrödinger equations)
34L05	General spectral theory of ordinary differential operators

Keywords:

Darboux transformation; Marchenko equation; Gel’fand-Levitan equation; Zakharov-Shabat system; nonlinear Schrödinger equation; integral equations; Fourier transformation; integral operator; ordinary differential operator; spectral problem; linear ordinary differential operator