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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The nonlinear Schrödinger equation on the half-line
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by Athanassios S. Fokas, A. Alexandrou Himonas and Dionyssios Mantzavinos PDF
Trans. Amer. Math. Soc. 369 (2017), 681-709 Request permission

Abstract:

The initial-boundary value problem (ibvp) for the cubic nonlinear Schrödinger (NLS) equation on the half-line with data in Sobolev spaces is analysed via the formula obtained through the unified transform method and a contraction mapping approach. First, the linear Schrödinger (LS) ibvp with initial and boundary data in Sobolev spaces is solved and the basic space and time estimates of the solution are derived. Then, the forced LS ibvp is solved for data in Sobolev spaces on the half-line $[0, \infty )$ for the spatial variable and on an interval $[0, T]$, $0<T<\infty$, for the temporal variable by decomposing it into a free ibvp and a forced ibvp with zero data, and its solution is estimated appropriately. Furthermore, using these estimates, well-posedness of the NLS ibvp with data $(u(x,0), u(0,t))$ in $H_x^s(0,\infty )\times H_t^{(2s+1)/4}(0,T)$, $s>1/2$, is established via a contraction mapping argument. In addition, this work places Fokas’ unified transform method for evolution equations into the broader Sobolev spaces framework.
References
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Additional Information
  • Athanassios S. Fokas
  • Affiliation: Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom
  • MR Author ID: 67825
  • Email: tf227@cam.ac.uk
  • A. Alexandrou Himonas
  • Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
  • MR Author ID: 86060
  • Email: himonas.1@nd.edu
  • Dionyssios Mantzavinos
  • Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
  • MR Author ID: 925372
  • Email: mantzavinos.1@nd.edu
  • Received by editor(s): July 17, 2014
  • Received by editor(s) in revised form: January 17, 2015
  • Published electronically: March 1, 2016
  • © Copyright 2016 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 681-709
  • MSC (2010): Primary 35Q55, 35G16, 35G31
  • DOI: https://doi.org/10.1090/tran/6734
  • MathSciNet review: 3557790