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Asymptotics for the multiple pole solutions of the nonlinear Schrödinger equation
Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics. Uniwersytet Jana Kochanowskiego Kielcach, Inst Matemat, Kielce, Poland.
2017 (English)In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 30, no 7, p. 2930-2981Article in journal (Refereed) Published
Abstract [en]

Multiple pole solutions consist of groups of weakly bound solitons. For the (focusing) nonlinear Schrodinger equation the double pole solution was constructed by Zakharov and Shabat. In the sequel particular cases have been discussed in the literature, but it has remained an open problem to understand multiple pole solutions in their full complexity.

In the present work this problem is solved, in the sense that a rigorous and complete asymptotic description of the multiple pole solutions is given. More precisely, the asymptotic paths of the solitons are determined and their position-and phase-shifts are computed explicitly. As a corollary we generalize the conservation law known for the N-solitons. In the special case of one wave packet, our result confirms a conjecture of Olmedilla.

Our method stems from an operator theoretic approach to integrable systems. To facilitate comparison with the literature, we also establish the link to the construction of multiple pole solutions via the inverse scattering method. The work is rounded off by many examples and MATHEMATICA plots and a detailed discussion of the transition to the next level of degeneracy.

Place, publisher, year, edition, pages
2017. Vol. 30, no 7, p. 2930-2981
Keyword [en]
nonlinear Schrodinger equation, multiple pole solutions, asymptotic behaviour, Cauchy-type determinants, inverse scattering method
National Category
Mathematics
Identifiers
URN: urn:nbn:se:miun:diva-31347DOI: 10.1088/1361-6544/aa6d9aISI: 000403576400002Scopus ID: 2-s2.0-85021249183OAI: oai:DiVA.org:miun-31347DiVA: diva2:1130204
Available from: 2017-08-08 Created: 2017-08-08 Last updated: 2017-08-10Bibliographically approved

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Schiebold, Cornelia

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CiteExportLink to record
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  • apa
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