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Graphene's honeycomb lattice structure is quite remarkable in the sense thatit leads, in the long wavelength limit, to a massless Dirac equationdescription of nonrelativistic quasiparticles associated with electrons andholes present in the two dimensional crystallite. In the case of cold bosonicatoms trapped in a honeycomb optical lattice, Haddad and Carr (2009) haverecently shown, by taking into account binary contact interactions, that thedynamics of these Bose-Einstein condensates is governed by a nonlinear Diracequation (NLDE). In this paper, we study exact stationary solutions of such aNLDE. After proving that the energy eigenvalues are real, we show that the sumof orbital angular momentum and pseudospin angular momentum normal to thecrystal commutes with the nonlinear Hamiltonian whenever magnitudes of thepseudospin components do not depend on the polar angle $\phi $. We obtain someexact stationary and localized solutions of the NLDE.

Some exact stationary state solutions of a nonlinear Dirac equation in 2+1 dimensions