group velocity of the locsiton, v$_{gr}$ = ( l a /T )( dδ/d q ), by taking a derivative of Eq. (16). Due to the anisotropy of Eq. (16), one may expect that the resulting dissipation and group velocity are highly dependent on the orientation of q with respect to E$_{in}$ . By its nature, the NRA only works well for long-wave locsitons, i. e., those with relatively small q . Indeed, for a long-wave locsiton, the LF at neighboring atoms differs insignificantly, so one may reasonably expect that replacing the neighboring dipoles at their actual positions with an effective dipole ring will not cause a significant change to the result. Our results obtained without resorting to the NRA (see Sec. III B), suggest that the NRA provides a good quantitative description of locsitons for q < ∼ π/ 2, while it gives reasonable qualitative estimates for q up to ∼ π . One may use the two first terms from the Taylor series expansion to analyze Eq. (16) at small q , which is easy to do by recalling that J$_{0}$ ( q ) = 1 − 1 $_{4}$q$^{2}$+ O ( q $^{4}$) and J$_{2}$ ( q ) = 1 $_{8}$q 2 + O ( q $^{4}$) [cf. Eq. (21.8-3) in Ref. [16]], so that In the long-wavelength limit, D NRA 2 ( q = 0 ) = 1 is reached at δ = δ$_{LL}$ , yielding the uniform, Lorentz, solution for the LF. The second term in the r. h. s. of Eq. (17) indicates that the anisotropy with respect to the locsiton polarization shows up even at very small q , where Eqs. (16) and (17) can be rewritten as Note that in describing locsitons at very small q (i. e., for δ ≈ δ$_{LL}$ ), the imaginary unit in the last formula cannot always be neglected compared to δ − δ$_{LL}$ . The locsiton dissipation in this case may become significant, depending on δ$_{LL}$ , and the necessity of considering complex q could appreciably complicate calculations. We will not go into the details of this special case in the present paper. In 1D arrays of atoms, the Lorentz resonance, δ = δ$_{LL}$ , always coincides with one of the edges of the locsiton frequency band [1, 2]. In that case, when one tunes the frequency of the incident laser beam towards ω$_{0}$ from the side of the Lorentz resonance, locsitons with the longest wavelength emerge first very close to the Lorentz resonance. The locsiton wavelength subsequently decreases as we approach and move past ω$_{0}$ . In a 2D lattice, the Lorentz resonance lies within the locsiton band, which means that, when similarly tuning the laser frequency towards ω$_{0}$ , short-wave locsitons will be excited first. In particular, D NRA 2 ( q ) ≈ 1 − 1 8 q $^{2}$[2 + 3 cos(2 ψ )] + O ( q $^{4}$) . (17) q 2 = − 8 2 + 3 cos(2 ψ ) ( δ − δ$_{LL}$ ) + i δ$_{LL}$ . (18) 13