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Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity

Borrelli, William (2017), Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity, Journal of Differential Equations, 263, 11, p. 7941-7964. 10.1016/j.jde.2017.08.029

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Type
Article accepté pour publication ou publié
Date
2017
Journal name
Journal of Differential Equations
Volume
263
Number
11
Publisher
Elsevier
Pages
7941-7964
Publication identifier
10.1016/j.jde.2017.08.029
Metadata
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Author(s)
Borrelli, William
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
In this paper we prove the existence of an exponentially localized stationary solution for a two-dimensional cubic Dirac equation. It appears as an effective equation in the description of nonlinear waves for some Condensed Matter (Bose-Einstein condensates) and Nonlinear Optics (optical fibers) systems. The nonlinearity is of Kerr-type, that is of the form |ψ| 2 ψ and thus not Lorenz-invariant. We solve compactness issues related to the critical Sobolev embedding H 1 2 (R 2 , C 2) → L 4 (R 2 , C 4) thanks to a particular radial ansatz. Our proof is then based on elementary dynamical systems arguments. Contents
Subjects / Keywords
Cubic Dirac equation; Dirac equation; Kerr nonlinearity; stationary solutions; graphene

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