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On the nonlinear Dirac equation on noncompact metric graphs

Borrelli, William; Carlone, Raffaele; Tentarelli, Lorenzo (2021), On the nonlinear Dirac equation on noncompact metric graphs, Journal of Differential Equations, 278, p. 326-357. 10.1016/j.jde.2021.01.005

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Type
Article accepté pour publication ou publié
Date
2021
Journal name
Journal of Differential Equations
Volume
278
Publisher
Elsevier
Published in
Paris
Pages
326-357
Publication identifier
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Author(s)
Borrelli, William
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Carlone, Raffaele
Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”
Tentarelli, Lorenzo
Dipartimento di Matematica "Guido Castelnuovo" [Roma I] [Sapienza University of Rome]
Abstract (EN)
The paper discusses the Nonlinear Dirac Equation with Kerr-type nonlinearity (i.e., ψp−2ψ) on noncompact metric graphs with a finite number of edges, in the case of Kirchhoff-type vertex conditions. Precisely, we prove local well-posedness for the associated Cauchy problem in the operator domain and, for infinite N-star graphs, the existence of standing waves bifurcating from the trivial solution at ω=mc2, for any p>2. In the Appendix we also discuss the nonrelativistic limit of the Dirac-Kirchhoff operator.
Subjects / Keywords
nonlinear Dirac equation; metric graphs; local well-posedness; bound states; implicit function theorem; bifurcation; perturbation method; nonrelativistic limit