also find here droplets which exist for $\beta =0$. Using Eqs. (\[pzero\]) and (\[e1d\]) we find their density corresponds to $\xi_0 \simeq 0.15$ which places such droplet far away from the quasi-1D regime and of course far away from the 3D system where such droplet cannot exist.

We now compare our predictions to the 1D results obtained in [@Petrov16]. To this end, we have to express the 3D interaction parameter by the 1D coupling, $g_\mathrm{1d}$. From Ref. [@olshanii], we infer that, for $a/L \ll 1$, the $g_\mathrm{1d}$ can be obtained by averaging the 3D interaction over the density profile in the tight directions, yielding $g_\mathrm{1d} = g/L^2 $. Using this relation, we obtain that in the quasi-1D regime the energy and equilibrium densities of the droplet have the same form as given in [@Petrov16; @dopiska1].

*Validity of the approach.*—We now briefly discuss the validity of our results. The Bogoliubov approach is valid as long as the LHY energy correction $\frac{\varepsilon_0}{L^3} e_\mathrm{LHY} $ is much smaller than the characteristic mean-field energy density $gn^2$ (for simplicity we take $g_{11}=g_{22}=g$). This condition reads $\frac{\pi L}{2 a} \xi^2 \gg |e_\mathrm{LHY}|$. For both situations analyzed in our paper, $e_\mathrm{LHY}^\mathrm{1d,2d}$ is practically equal to $e_\mathrm{LHY}^\mathrm{3d}$ for 