## Wednesday, May 29, 2013

We present a thorough analysis of soliton solutions to the quasi-one-dimensional zigzag and armchair nonlinear Dirac equation (NLDE) for a Bose-Einstein condensate in a honeycomb lattice. The two types of NLDEs correspond to quasi-one-dimensional reductions in two independent directions, in direct analogy to the narrowest of graphene nanoribbons. We analyze the solution space of the quasi-one-dimensional NLDE by finding fixed points, delineating the various regions in solution space, and through an invariance relation which we obtain as a first integral of the NLDE. For both the zigzag and armchair geometries we obtain spatially oscillating multi-soliton as well as asymptotically flat single soliton solutions using five different methods: by direct integration; an invariance relation; parametric transformation; a series expansion; and by numerical shooting. By tuning the ratio of the chemical potential to the nonlinearity for a fixed value of the Dirac spinor kinetic energy, we can obtain both bright and dark solitons over a nonzero background. The density contrast between the dark (bright) soliton notch (peak) and the background is about 2/3 (3/1). For both soliton types, we compute the discrete spectra for several spatially quantized states in a harmonic potential. We interpret our solitons as topologically protected domain walls in a quasi-one-dimensional system which separate distinct regions of pseudospin-$1/2$ with $S_z = \pm 1/2$. By solving the relativistic linear stability equations we obtain the low-energy spectrum for excitations in the bulk region far from the soliton core and for bound states in the core and find that excitations occur as spin waves and as a Nambu-Goldstone mode. For a Bose-Einstein condensate of $^{87}\mathrm{Rb}$ atoms, we find that our soliton solutions are stable on time scales relevant to experiments.