A Zakharov-Shabat inverse scattering problem with a polynomial spectral dependence in the potential
type: Item
instance of: scholarly article
title: A Zakharov-Shabat inverse scattering problem with a polynomial spectral dependence in the potential
zbMATH Open document ID: 0558.34021
DOI: 10.1007/BF00403271
author: M. O. Guillaume
author: Marcel Jaulent
published in: Letters in Mathematical Physics
publication date: 1982-01-01T00:00:00Z
review text: The inverse scattering problem for the Zakharov-Shabat system \(y'+iK\sigma_ 3y=vy\), \(x\in {\mathbb{R}}\), \(y=\left( \begin{matrix} y_ 1\\ y_ 2\end{matrix} \right)\) is studied, where the 'potential' v depends on the spectral parameter \(K: v(K,x)=\sum^{n}_{p=0}K^{p/2n}v_ p(x),\) \(v_ p=\left( \begin{matrix} 0\\ u^-_ p\end{matrix}^{u^+_ p}_{0}\right)\). The \(2(n+1)\) 'scalar potentials' \(u^-_ p\) and \(u^-_ p\) are supposed to be sufficiently regular complex functions of \(x\in {\mathbb{R}}\), decreasing fast enough as \(| x| \to \infty.\) The problem is solved by reduction to the inverse scattering problem for two coupled Zakharov-Shabat systems in 4n-dimensional space \(y^{\pm '}+iK(\sigma_ 3\otimes I_{2n})y^{\pm}=(U^{\pm}\pm KQ^{\pm})y^{\pm}.\)
Mathematics Subject Classification ID: 34L99
Mathematics Subject Classification ID: 34A55
zbMATH DE Number: 3889932
zbMATH Keywords: inverse scattering problem
zbMATH Keywords: Zakharov-Shabat system
reviewed by: Joachim Weidmann
MaRDI profile type: Publication
cites work: An exact solution for a derivative nonlinear Schrödinger equation
cites work: A Schrödinger inverse scattering problem with a spectral dependence in the potential
cites work: Nonlinear evolution equations associated with energy-dependent Schrödinger potentials
cites work: Schrödinger spectral problems with energy–dependent potentials as sources of nonlinear Hamiltonian evolution equations
description: scientific article; zbMATH DE number 3889932