On equations solvable by the inverse scattering method for the Dirac operator
type: Item
instance of: scholarly article
title: On equations solvable by the inverse scattering method for the Dirac operator
zbMATH Open document ID: 1028.37042
DOI: 10.1016/S1007-5704(02)00011-4
author: V. K. Mel'nikov
published in: Communications in Nonlinear Science and Numerical Simulation
publication date: 2003-04-03T00:00:00Z
review text: The paper aims to explore the characteristic property of Lax representations of integrable evolution equations. For the so-called Dirac operator, which leads to the AKNS spectral problem, the author proves that the \(S\)-matrix (or the scattering data) of the Dirac operator under an evolution equation satisfies a regular first-order ordinary differential equation, with an independent variable being time, if and only if the evolution equation possesses a Lax representation associated with the Dirac operator. Such Lax representations are equivalent to zero curvature representations and need the isospectral condition. The technique developed is constructive, which shows the relation between Lax representations and the governing equations for the scattering data. It would be also possible to apply the author's technique of using the \(S\)-matrix to characterize integrable evolution equations associated with other spectral problems and even under nonisospectral conditions.
reviewed by: Wen-Xiu Ma
Mathematics Subject Classification ID: 37K15
Mathematics Subject Classification ID: 35Q51
Mathematics Subject Classification ID: 37K40
zbMATH DE Number: 1892195
zbMATH Keywords: integrable evolution equations
zbMATH Keywords: inverse scattering method
zbMATH Keywords: Dirac operator
zbMATH Keywords: \(S\)-matrix
zbMATH Keywords: Lax representation
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1892195