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Perturbative and exact results on the neumann value for the nonlinear Schrödinger on the half-line
2014 (English)In: Journal of Physics, Conference Series, ISSN 1742-6588, E-ISSN 1742-6596, Vol. 482, no 1, 012015Article in journal (Refereed) Published
Abstract [en]

The most challenging problem in the implementation of the so-called unified transform to the analysis of the nonlinear Schrödinger equation on the half-line is the characterization of the unknown boundary value in terms of the given initial and boundary conditions. For the so-called linearizable boundary conditions this problem can be solved explicitly. Furthermore, for non-linearizable boundary conditions which decay for large t, this problem can be largely bypassed in the sense that the unified transform yields useful asymptotic information for the large t behavior of the solution. However, for the physically important case of periodic boundary conditions it is necessary to characterize the unknown boundary value. Here, we first present a perturbative scheme which can be used to compute explicitly the asymptotic form of the Neumann boundary value in terms of the given τ-periodic Dirichlet datum to any given order in a perturbation expansion. We then discuss briefly an extension of the pioneering results of Boutet de Monvel and co-authors which suggests that if the Dirichlet datum belongs to a large class of particular τ-periodic functions, which includes {a exp(iωt)|a > 0, ω ≥ a2}, then the large t behavior of the Neumann value is given by a τ-periodic function which can be computed explicitly.

Place, publisher, year, edition, pages
2014. Vol. 482, no 1, 012015
Keyword [en]
Boundary conditions, Nonlinear equations, Asymptotic forms, Dinger equation, Initial and boundary conditions, Neumann boundary, Periodic boundary conditions, Periodic function, Perturbation expansions, Unknown boundary, Mathematical transformations
National Category
URN: urn:nbn:se:kth:diva-163793DOI: 10.1088/1742-6596/482/1/012015ISI: 000334352400014ScopusID: 2-s2.0-84896921189OAI: diva2:802399
Conference on Physics and Mathematics of Nonlinear Phenomena (PMNP), 22 June 2013 through 29 June 2013, Gallipoli

QC 20150417

Available from: 2015-04-13 Created: 2015-04-12 Last updated: 2015-04-17Bibliographically approved

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Lenells, Jonatan
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