Spinor solitons in one-dimensional and planar nonlinear Dirac equations

• Gap solitons. • Nonlinear vortex solutions. • Numerical results. • Analytical results. • Linear stability analysis. This article investigates nonlinear Dirac equations (NLD) with cubic-type nonlinearities, that is, vector and scalar self-interaction nonlinearities. We present analytical solutions of gap-solitons, which are self-localised, moving or quiescent pulses existing in the band gaps of nonlinear Dirac models. We also perform the linear stability analysis of the gap-soliton bearing systems and find that the gap-solitons possess some regions of instability. We also extend our studies to planar nonlinear Dirac equations and relate them to light propagation in photonic lattices, such as photonic graphene and present their numerical solutions, in particular, the rotationally symmetric localised radial profiles that rotate about the Brillouin zone.