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Novel formulation of inverse scattering and characterization of scattering data

Pages: 343 - 350, Issue Special, September 2011

 Abstract        Full Text (305.1K)              

Francesco Demontis - Dip. Matematica e Informatica, Università di Cagliari, Viale Merello 92, 09121 Cagliari, Italy (email)
Cornelis Van der Mee - Dip. Matematica e Informatica, Università di Cagliari, Viale Merello 92, 09121 Cagliari, Italy (email)

Abstract: In this article we formulate the direct and inverse scattering theory for the focusing matrix Zakharov-Shabat system as the construction of a 1, 1-correspondence between focusing potentials with entries in $L^1(\mathbb{R})$ and Marchenko integral kernels, given the fact that these kernels encode the usual scattering data (one reflection coecient, the discrete eigenvalues with positive imaginary part, and the corresponding norming constants) faithfully. In the re ectionless case, we solve the Marchenko equations explicitly using matrix triplets and obtain focusing matrix NLS solutions in closed form.

Keywords:  Inverse Scattering Transform, Marchenko equation, Characterization of Scattering Data
Mathematics Subject Classification:  Primary: 35Q55, 37K15

Received: July 2010;      Revised: January 2011;      Published: October 2011.