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Article Dans Une Revue Journal of Differential Equations Année : 2017

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

Résumé

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
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Dates et versions

hal-01540930 , version 1 (16-06-2017)
hal-01540930 , version 2 (28-06-2017)

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William Borrelli. Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity. Journal of Differential Equations, 2017. ⟨hal-01540930v2⟩
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