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

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 |$\psi$| 2 $\psi$ and thus not Lorenz-invariant. We solve compactness issues related to the critical Sobolev embedding H 1 2 (R 2 , C 2) $\to$ L 4 (R 2 , C 4) thanks to a particular radial ansatz. Our proof is then based on elementary dynamical systems arguments. Contents

References in corpus (4)

• The electronic properties of graphene
• Dirac materials
• Introduction to Dirac materials and topological insulators
• Stability of solitary waves and vortices in a 2D nonlinear Dirac model

Cited by in corpus (8)

• Nonlinear Dirac Equation On Graphs With Localized Nonlinearities: Bound States And Nonrelativistic Limit
• On the nonlinear Dirac equation on noncompact metric graphs
• A note on the Dirac operator with Kirchoff-type vertex conditions on noncompact metric graphs
• Multiple solutions for a self-consistent Dirac equation in two dimensions
• An overview on the standing waves of nonlinear Schrödinger and Dirac equations on metric graphs with localized nonlinearity
• Ground state Dirac bubbles and Killing spinors
• Some properties of Dirac-Einstein bubbles
• Curvature effect in the spinorial Yamabe problem on product manifolds