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The nonlinear Schrödinger equation on the half-line with a Robin boundary condition

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Abstract

The initial-boundary value problem for the nonlinear Schrödinger equation on the half-line with initial data in Sobolev spaces \(H^s(0, \infty )\), \(1/2< s\leqslant 5/2\), \(s\ne 3/2\), and Robin boundary data of appropriate regularity is shown to be locally well-posed in the sense of Hadamard. The proof is through a contraction mapping argument and hence relies crucially on certain estimates for the forced linear counterpart of the nonlinear problem. In particular, the essence of the analysis lies in the pure linear initial-boundary value problem, which corresponds to the case of zero forcing, zero initial data, and nonzero boundary data. This problem, which is studied by taking advantage of the solution formula derived via the unified transform of Fokas, holds an instrumental role in the overall analysis as it reveals the correct function space for the Robin boundary data.

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Acknowledgements

This work was partially supported by a grant from the Simons Foundation (#524469 to Alex Himonas). Both authors are thankful to the anonymous reviewer of the manuscript for constructive remarks and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556, USA

    A. Alexandrou Himonas

  2. Department of Mathematics, University of Kansas, Lawrence, KS, 66045, USA

    Dionyssios Mantzavinos

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  1. A. Alexandrou Himonas
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  2. Dionyssios Mantzavinos
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Correspondence to Dionyssios Mantzavinos.

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This work was partially supported by a grant from the Simons Foundation (#524469 to Alex Himonas).

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Himonas, A.A., Mantzavinos, D. The nonlinear Schrödinger equation on the half-line with a Robin boundary condition. Anal.Math.Phys. 11, 157 (2021). https://doi.org/10.1007/s13324-021-00589-y

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