Exact localized and oscillatory solutions of the nonlinear spin and pseudospin symmetric Dirac equations

We derive exact analytic solutions of nonlinear Dirac equations with Kerr-like cubic nonlinearity. The equations model, among many physical systems, a Bose-Einstein condensate confined by a two-dimensional honeycomb optical lattice as well as relativistic solitons in optical waveguide arrays. The nonrelativistic limit of the solutions is derived, and the role of the nonlinearity on localization is discussed. The possibility of realizing some of these solutions in waveguide arrays and Bose-Einstein condensates is pointed out.