Weakly localized states for nonlinear Dirac equations

TL;DR

This paper proves the existence of infinitely many weakly localized, non-square-integrable stationary solutions for massless Dirac equations in 2D, relevant to honeycomb structures, with explicit asymptotics and variational characterization.

Contribution

It introduces a direct radial ansatz method to establish solutions and extends previous massive case results to more general nonlinearities.

Findings

Existence of infinitely many solutions with specific asymptotics

Solutions are non-square-integrable and weakly localized

Extension of previous results to broader nonlinearities

Abstract

We prove the existence of infinitely many non square-integrable stationary solutions for a family of massless Dirac equations in 2D. They appear as effective equations in two dimensional honeycomb structures. We give a direct existence proof thanks to a particular radial ansatz, which also allows to provide the exact asymptotic behavior of spinor components. Moreover, those solutions admit a variational characterization. We also indicate how the content of the present paper allows to extend our previous results for the massive case [5] to more general nonlinearities.