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.

Figure 1

The exact stationary solutions (13, 14, 15, 16), (26), (25), (32), and (33) of the NLD Eqs. (3) and (4). (a) Nonrelativistic spin symmetric case with $m=-{\alpha }_{+}=1$ and $x_0=0$. (b) Nonrelativistic pseudospin symmetric case with $m={\alpha }_{-}/4=1/4$ and $c_1=x_0=0$. (c) Relativistic spin symmetric case with $\lambda =10m=-{\alpha }_{+}=1$ and $x_0=0$. (d) Relativistic pseudospin symmetric case with $\lambda =1$, ${\alpha }_{-}=0.8$, $m=1.2$, and $x_0=0$. The solid blue line corresponds to ${\varphi }_{+}$ and the dashed red line corresponds to ${\varphi }_{-}$.

Figure 2

The exact oscillatory solutions (40) and (41) of the NLD Eqs. (28) and (29) for the relativistic pseudospin symmetric case with $m=-{\alpha }_{-}=1$, $\lambda =5$, and $c_1=x_0=0$. The solid blue line corresponds to ${\varphi }_{+}$ and the dashed red line corresponds to ${\varphi }_{-}$.

Figure 3

Total spin $\overline{Q}$ and total energy $\overline{E}$ vs $\lambda /m$. The upper and lower dashed horizontal lines indicate the nonrelativistic and massless values [$\sqrt{8}\pi /{\alpha }_{+}$ and $3\pi /(\sqrt{8}/{\alpha }_{+})$], respectively. The value ${\alpha }_{+}=-1$ is used.