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

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Abstract

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|2ψ and thus not Lorenz-invariant. We solve compactness issues related to the critical Sobolev embedding H[Formula presented](R2,C2)↪L4(R2,C4) thanks to a particular radial ansatz. Our proof is then based on elementary dynamical systems arguments.
Original languageEnglish
Pages (from-to)7941-7964
Number of pages24
JournalJournal of Differential Equations
Volume263
DOIs
Publication statusPublished - 2017

Keywords

  • critical Dirac equations

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