the order parameter is complex, it can be expressed as $m(\textbf{r})=m(\rho)\text{e}^{i(\sigma\theta+kz)}$ where the amplitude $m(\rho)$ takes care of inhomogeneity of the condensate density ($n_0(\textbf{r})=|m(\textbf{r})|^2$) along the radial direction and the phase $\sigma\theta+kz$ accounts for the flow of the condensate either rotationally along the $\hat{\theta}$ direction or translationally along the $\hat{z}$ direction. For simplicity, we consider the condensate to be static inside the cylinder so that the condensate order parameter can be written as a real function of the radial coordinate as $m(\textbf{r})=m(\rho)$. We can express the Ginzburg-Landau form in Eq.  in terms of the BEC order parameter $m(\rho)$ as, $$\begin{aligned}
\label{eqn:mf1}
\Omega_0=\frac{\hbar^2N_0}{M}\int\bigg[\frac{1}{2}[\nabla m(\rho)]^2-\frac{|a|}{2}m^2(\rho)+\frac{b}{4}m^4(\rho)\bigg]\text{d}^3\textbf{r},~\end{aligned}$$ where we have defined, ${a} = \frac{2M\mu}{\hbar^2}=\frac{2Mg\bar{n}}{\hbar^2}$ and $ b = \frac{2MgN_0}{\hbar^2}>0$. The minimization of the energy in Eq.  with respect to the order parameter $m(\rho)$, [*i.e. *]{}$ \frac{\delta\Omega_0}{\delta m}=0$, leads to the following equation for the profile of the order parameter, $$\label{eqn:39}
-\nabla^2m-|a|m+bm^3=0~,$$ which can be rewritten in terms of the cylindrical polar coordinates as $$\label{eqn:40}
-\frac{\text{d}^2m}{\text{d}\rho^2}-\frac{1}{\rho}\frac{\text{d}m}{\text{d}\rho}-|a|m+bm^3=0~.$$ Note that, for a large system, the order parameter is expected to be homogeneous, and in the bulk limit we have $m \to m_0=\sqrt{|a|/b}$.

The mean-field component of the force is given by $F_R^{(\text{mf})}=-\frac{\delta \Omega_0}{\delta R}$. It is not 