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

TL;DR

This paper analyzes soliton solutions of a nonlinear Dirac equation modeling Bose-Einstein condensates in honeycomb optical lattices, revealing various soliton types and solution behaviors through analytical and numerical methods.

Contribution

It provides the first comprehensive analysis of soliton solutions in the nonlinear Dirac equation for BECs in armchair nanoribbon lattices, including new solution methods and classifications.

Findings

Existence of bright and dark solitons in the system.

Multiple analytical and numerical methods successfully find soliton solutions.

Identification of fixed points and solution space regions in the NLDE.

Abstract

We present a thorough analysis of soliton solutions to the quasi-one-dimensional nonlinear Dirac equation (NLDE) for a Bose-Einstein condensate in a honeycomb lattice with armchair geometry. Our NLDE corresponds to a quasi-one-dimensional 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-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. 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.