Closure of the Manakov System

We discuss the closure of the product eigenstate for the Manakov system. Although the analysis is similar in principle to that for the Zakharov--Shabat system published elsewhere, two additional features arise which require careful attention. First, in addition to the direct (or forward) scattering...

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Bibliographic Details
Published inSIAM journal on mathematical analysis Vol. 32; no. 1; pp. 54 - 79
Main Authors Docksey, R. G., Elgin, J. N.
Format Journal Article
LanguageEnglish
Published Philadelphia Society for Industrial and Applied Mathematics 2000
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Summary:We discuss the closure of the product eigenstate for the Manakov system. Although the analysis is similar in principle to that for the Zakharov--Shabat system published elsewhere, two additional features arise which require careful attention. First, in addition to the direct (or forward) scattering problem, an adjoint scattering problem is necessary. In the Zakharov--Shabat system, the adjoint problem is trivially related to the direct problem, leading to the formation of "squared" eigenstates rather than the product eigenstates which we derive. Second, the system is not diagonal in the sense that both parts of the potential q1 and q2 contribute to each element Sij of the scattering data, while---reciprocally---all (relevant) elements of the scattering data contribute in the reconstruction of both q1 and q2.
ISSN:0036-1410
1095-7154
DOI:10.1137/S0036141098343677