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dc.contributor.authorAlkın, Aykut-
dc.contributor.authorMantzavinos, Dionyssios-
dc.contributor.authorÖzsarı, Türker-
dc.description.abstractWe establish local well-posedness in the sense of Hadamard for a certain third-order nonlinear Schrödinger equation with a multiterm linear part and a general power nonlinearity, known as higher-order nonlinear Schrödinger equation, formulated on the half-line (Formula presented.). We consider the scenario of associated coefficients such that only one boundary condition is required and hence assume a general nonhomogeneous boundary datum of Dirichlet type at (Formula presented.). Our functional framework centers around fractional Sobolev spaces (Formula presented.) with respect to the spatial variable. We treat both high regularity ((Formula presented.)) and low regularity ((Formula presented.)) solutions: in the former setting, the relevant nonlinearity can be handled via the Banach algebra property; in the latter setting, however, this is no longer the case and, instead, delicate Strichartz estimates must be established. This task is especially challenging in the framework of nonhomogeneous initial-boundary value problems, as it involves proving boundary-type Strichartz estimates that are not common in the study of Cauchy (initial value) problems. The linear analysis, which forms the core of this work, crucially relies on a weak solution formulation defined through the novel solution formulae obtained via the Fokas method (also known as the unified transform) for the associated forced linear problem. In this connection, we note that the higher-order Schrödinger equation comes with an increased level of difficulty due to the presence of more than one spatial derivatives in the linear part of the equation. This feature manifests itself via several complications throughout the analysis, including (i) analyticity issues related to complex square roots, which require careful treatment of branch cuts and deformations of integration contours; (ii) singularities that emerge upon changes of variables in the Fourier analysis arguments; and (iii) complicated oscillatory kernels in the weak solution formula for the linear initial-boundary value problem, which require a subtle analysis of the dispersion in terms of the regularity of the boundary data. The present work provides a first, complete treatment via the Fokas method of a nonhomogeneous initial-boundary value problem for a partial differential equation associated with a multiterm linear differential operator. © 2023 Wiley Periodicals LLC.en_US
dc.description.sponsorshipThe authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, for support and hospitality during the program “Dispersive Hydrodynamics,” when work on this paper was undertaken (EPSRC Grant Number EP/R014604/1). The second author gratefully acknowledges partial support from the U.S. National Science Foundation (NSF‐DMS 2206270).en_US
dc.relation.ispartofStudies in Applied Mathematicsen_US
dc.subjectFokas methoden_US
dc.subjectSchrödinger equationen_US
dc.subjectInitial-boundary value problemsen_US
dc.subjectKorteweg–de Vries equationen_US
dc.subjectPower nonlinearityen_US
dc.subjectStrichartz estimatesen_US
dc.subjectInitial value problemsen_US
dc.subjectSobolev spacesen_US
dc.titleLocal well-posedness of the higher-order nonlinear Schrödinger equation on the half-line: Single-boundary condition caseen_US
dc.institutionauthorAlkın, Aykut-
dc.departmentİzmir Institute of Technology. Mathematicsen_US
dc.relation.publicationcategoryMakale - Uluslararası Hakemli Dergi - Kurum Öğretim Elemanıen_US
item.fulltextWith Fulltext-
item.openairetypeArticle- Department of Mathematics-
Appears in Collections:Mathematics / Matematik
Scopus İndeksli Yayınlar Koleksiyonu / Scopus Indexed Publications Collection
WoS İndeksli Yayınlar Koleksiyonu / WoS Indexed Publications Collection
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