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

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 H12(R2,C2)↪L4(R2,C4) thanks to a particular radial ansatz. Our proof is then based on elementary dynamical systems arguments.