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The nonlinear Schrödinger equation on the half-line


Authors: Athanassios S. Fokas, A. Alexandrou Himonas and Dionyssios Mantzavinos
Journal: Trans. Amer. Math. Soc. 369 (2017), 681-709
MSC (2010): Primary 35Q55, 35G16, 35G31
DOI: https://doi.org/10.1090/tran/6734
Published electronically: March 1, 2016
MathSciNet review: 3557790
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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.


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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
Email: tf227@cam.ac.uk

A. Alexandrou Himonas
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: himonas.1@nd.edu

Dionyssios Mantzavinos
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: mantzavinos.1@nd.edu

DOI: https://doi.org/10.1090/tran/6734
Keywords: Nonlinear Schr\"odinger equation, integrable, solitons, unified transform method, initial-boundary value problem, well-posedness, Sobolev spaces, space and time estimates, Laplace, contraction mapping.
Received by editor(s): July 17, 2014
Received by editor(s) in revised form: January 17, 2015
Published electronically: March 1, 2016
Article copyright: © Copyright 2016 American Mathematical Society