## 4 Stationary Solutions
### Case \(g\
eq 0\)  
For the solutions with non-zero values of \(g\), we propose a generalization of the empirical density profile given by  
\[\rho_{\text{core}}(r)=\rho_{c}\left[1+\left(2^{1/8}-1\right)\left(\frac{r}{r_{c }}\right)^{2+\beta}\right]^{-8}, \tag{24}\]  
where \(r_{c}=r_{c}(\rho_{c},g)\) and \(\beta=\beta(\rho_{c},g)\) determine the core radius and exponent, respectively. The total mass of the core can be integrated to give  
\[M_{\text{core}} = \int_{0}^{\infty}\rho_{\text{core}}(r)r^{2}dr \tag{25}\] \[= \frac{(2^{1/8}-1)^{-\frac{3}{2+\beta}}}{5040}\rho_{c}r_{c}^{3} \frac{\Gamma\left(\frac{3}{2+\beta}\right)\Gamma\left(8-\frac{3}{2+\beta} \right)}{2+\beta}\] \[= M(r_{\text{max}}).\]  
Solving for the core radius \(r_{c}\) one obtains a closed expression:  
\[r_{c}=\left[\frac{5040(2+\beta)M_{\text{core}}}{\left(2^{1/8}-1\right)^{-\frac{ 3}{2+\beta}}\Gamma\left(\frac{3}{2+\beta}\right)\Gamma\left(8-\frac{3}{2+ \beta}\right)}\right]^{1/3}. \tag{26}\]  
In the isolated case, where no atmosphere of bosons is around the core like in BEC dark-matter collapse simulations, the core mass is also the total mass from the numerical solution \(M_{\text{core}}=M(r_{\text{max}})\), and \(\beta\) is the only free parameter of \(\rho_{\text{core}}(r)\). Consequently, \(\beta\) is the only fitting parameter to match the ansatz (24) with the numerical solution of the eigenvalue problem. The optimal \(\beta\) value is found via a genetic algorithm using \(\mathbf{DNA}=(\beta)\) and the fitness function \(f(\beta)=\left[\int_{0}^{r_{\text{max}}}|\rho_{\text{core}}-\psi^{2}|\right]^{ -1}dr\), with a tolerance of \(10^{6}\) for the best individual. We propose an empirical function, similar to (26) but for \(g\
eq 0\), for the core radius and exponent, subject to the following constraints:  
1. According to the \(\lambda\)-scaling, \(r_{c}\sim\rho_{c}^{-1/4}\) and \(r_{c}\sim g^{1/2}\). 2. \(\beta\) should be \(\lambda\)-invariant. 3. When \(g=0,r_{c}=1.306\rho_{c}^{-1/4}\) and \(\beta=0\) are recovered. Figure 1: On the left we show the inverse of the best individual’s fitness as a function of generations for the case \((\psi_{c},g)=(1,0)\), demonstrating that the GA method provides solutions approaching the satisfaction of stationary GPP equations very rapidly. On the right, we compare the solution found by the GA with empirical Formula (22), for the case with \(\rho_{c}=1\) and \(r_{c}\approx 1.306\). The transformation (3) implies that the coefficient \(\alpha=g\rho_{c}^{1/2}\) is \(\lambda\)-invariant; therefore, \(\beta\) should only depend on \(\alpha\). Based on these conditions, the following functions are proposed:  
\[r_{c} = 1.306\rho_{c}^{-1/4}(1+a_{1}\alpha+a_{2}\alpha^{2}), \tag{27}\] \[\beta = b_{1}\alpha+b_{2}\alpha^{2}+b_{3}|\alpha|^{1/2}. \tag{28}\]  
where the fitting parameters are \(a_{1}=0.3681\pm 24.50\times 10^{-5}\), \(a_{2}=0.0905\pm 31.34\times 10^{-5}\), \(b_{1}=0.2842\pm 10.71\times 10^{-5}\), \(b_{2}=0.0845\pm 21.80\times 10^{-5}\), and \(b_{3}=-0.0117\pm 5.443\times 10^{-5}\). To see that the general empirical formula is a good approximation to the fundamental state, we show in Figure 2 a plot of \(r_{c}\) as a function of \(\rho_{c}\) in Equation (27) for different values of \(g\), both using the solution of the eigenvalue problem and empirical Formula (26). Notice that \(r_{c}\) decreases with increasing \(\rho_{c}\) and increases with increasing \(g\) as expected. The plots show how good the empirical formula resembles the properties of the solution to the eigenvalue problem. We also show \(\beta\) vs. \(r_{c}\) implicit from Equations (27) and (28). Additionally, \(\beta\) changes sign when \(g\) changes sign. There is an empirical formula in [20], similar to (24), constructed based on simulations of core-formation simulations.