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Symmetry 2019, 11(2), 169; https://doi.org/10.3390/sym11020169

An Overview on the Standing Waves of Nonlinear Schrödinger and Dirac Equations on Metric Graphs with Localized Nonlinearity

1
Université Paris-Dauphine, PSL Research University, CNRS, UMR 7534, CEREMADE, F-75016 Paris, France
2
Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, MSA, via Cinthia, I-80126 Napoli, Italy
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Received: 9 January 2019 / Revised: 25 January 2019 / Accepted: 27 January 2019 / Published: 1 February 2019
(This article belongs to the Special Issue Symmetries of Nonlinear PDEs on Metric Graphs and Branched Networks)
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Abstract

We present a brief overview of the existence/nonexistence of standing waves for the NonLinear Schrödinger and the NonLinear Dirac Equations (NLSE/NLDE) on metric graphs with localized nonlinearity. First, we focus on the NLSE (both in the subcritical and the critical case) and, then, on the NLDE highlighting similarities and differences with the NLSE. Finally, we show how the two equations are related in the nonrelativistic limit by the convergence of the bound states. View Full-Text
Keywords: metric graphs; NLS; NLD; ground states; bound states; localized nonlinearity; nonrelativistic limit metric graphs; NLS; NLD; ground states; bound states; localized nonlinearity; nonrelativistic limit
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Borrelli, W.; Carlone, R.; Tentarelli, L. An Overview on the Standing Waves of Nonlinear Schrödinger and Dirac Equations on Metric Graphs with Localized Nonlinearity. Symmetry 2019, 11, 169.

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