Figure 3. Comparison of the variational (red lines) and 3D eGPE (black lines) solutions for a uniform infinite system. The 1/e density contours of the transverse modes of the 3D eGPE χ and the variational approach χσ for (a) ${a}_{s}=120{a}_{0}$ and (b) ${a}_{s}=95{a}_{0}$. The harmonic oscillator ground state ${\chi }_{\mathrm{ho}}$ is shown for reference (blue lines). In (c) and (d) we compare the transverse mode profiles along the x (dash–dot) and y (lines) axes for the cases given in (a) and (b), respectively. (e) The effective 1D k-space interaction kernel obtained from the various transverse functions for ${a}_{s}=120{a}_{0}$ (dashed lines) and ${a}_{s}=95{a}_{0}$ (solid lines). The 3D eGPE result ${\tilde{U}}_{z}$ is obtained by evaluating equation (13) using χ. The variational ${\tilde{U}}_{\sigma }$ and the harmonic oscillator ${\tilde{U}}_{\mathrm{ho}}$ results are obtained from equation (18). Results for 164Dy using ${a}_{{dd}}=130.8\,{a}_{0}$, with ${\omega }_{x,y}=2\pi \times 150\,\mathrm{Hz}$, ${\omega }_{z}=0$, and $n=2.5\times {10}^{3}\ \mu {{\rm{m}}}^{-1}$.
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