*Published paper*

**Inserted:** 15 sep 2017

**Last Updated:** 29 sep 2017

**Journal:** Journal of Differential Equations

**Year:** 2017

**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 bers) systems. The nonlinearity is of Kerr-type, that is of the form $\vert\psi\vert^{2}\psi$ and thus not Lorenz-invariant. We solve compactness issues related to the critical Sobolev embedding $H^{\frac{1}{2}}(\mathbb{R}^{2};\mathbb{C}^{2})\hookrightarrow L^{4}(\mathbb{R}^{2};\mathbb{C}^{2})$ thanks to a particular radial ansatz. Our proof is then based on elementary dynamical systems arguments.

**Keywords:**
cubic Dirac equation, graphene, shooting method, nonlinear waves