The nonlinear Dirac equation in Bose–Einstein condensates: I. Relativistic solitons in armchair nanoribbon optical lattices

Abstract

We present a thorough analysis of soliton solutions to the quasi-one-dimensional (quasi-1D) nonlinear Dirac equation (NLDE) for a Bose-Einstein condensate in a honeycomb lattice with armchair geometry. Our NLDE corresponds to a quasi-1D reduction of the honeycomb lattice along the zigzag direction, in direct analogy to graphene nanoribbons. Excitations in the remaining large direction of the lattice correspond to the linear subbands in the armchair nanoribbon spectrum. Analytical as well as numerical soliton Dirac spinor solutions are obtained. We analyze the solution space of the quasi-1D 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. We obtain spatially oscillating multi-soliton solutions 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 energy-momentum tensor, we can obtain both bright and dark solitons over a nonzero density background.

The nonlinear Dirac equation in Bose–Einstein condensates: II. Relativistic soliton stability analysis

“…for D x , with an overall complex constant prefactor omitted for clarity. As we showed in [25], NLDE solitons interpolate between two asymptotic states that are linear combinations of ( ) ( )…”

“…x ω ω ≫ , as we discussed in [25] with the soliton in the x-direction, we encounter only the effects resulting from quantization of spatial modes along the soliton direction shown in figure 1. We then take the longitudinal trapping potential to be V x M x ( )…”

“…Note that now equations (23) and (24) are not coupled to equations (25) and (26). Next, we make the substitution…”

“…and eigenvalues E k . The parameters in equations (12)-(15) are already renormalized due to dimensional reduction from 2D to quasi-1D as described in Section 3 of [25]. We point out that equations (12)-(15) pertain to the NLDE associated with the real Dirac operator.…”

Remarks on Nonlinear Dirac Equations in One Space Dimension

“…The system (1.5) does not have its own name in physical literature but the following system has been considered in [6] where they present an analysis of soliton solutions to the following quasi-one dimensional Dirac system for a Bose-Einstein condensate.…”