## 2 Model and Equations  
In a Bose-Einstein Condensate (BEC), a significant portion of the particles occupy the lowest quantum state, resulting in overlapping wave functions. Consequently, the state of a BEC can be effectively described by a collective wave function, known as the order parameter, \(\Psi(t,\vec{x})\). Due to interactions between bosons, which induce nonlinear effects, this collective wave function satisfies the Gross-Pitaevskii (GP) equation:  
\[\mathrm{i}\hbar\frac{\partial\Psi}{\partial t}=\left[-\frac{\hbar^{2}}{2m} \
abla^{2}+mV+g|\Psi|^{2}\right]\Psi, \tag{1}\]  
where \(\hbar\) is the reduced Planck constant, \(m\) is the mass of the boson, \(V\) is the external potential acting as a gas trap, and \(g=\frac{4\pi\hbar^{2}a_{s}}{m}\) is the nonlinear coefficient, where \(a_{s}\) is the scattering length of two interacting bosons. The concept of BEC Dark Matter (BEC-DM) hypothesizes that dark matter exists in the form of a BEC, where the trapping potential is self-generated by the bosonic ensemble through the Poisson equation:  
\[\
abla^{2}V=4\pi Gm|\Psi|^{2}. \tag{2}\]  
The nonlinear system (1) and (2) is known as the Gross-Pitaevskii-Poisson (GPP) system and exhibits scale invariance, described by the transformation [13]:  
\[\{t,\vec{x},\Psi,V,g\}\rightarrow\left\{\lambda^{-2}t,\lambda^{-1}\vec{x}, \lambda^{2}\Psi,\lambda^{2}V,\lambda^{-2}g\right\}, \tag{3}\]  
where \(\lambda\) is a scaling factor. An alternative description of the GPP system (1) and (2) can be obtained through the Madelung transformation \(\Psi=\sqrt{\rho/m}e^{iS/\hbar}\), where \(\rho=m|\Psi|^{2}\) is the mass density, and \(S\) represents the phase of the wave function. By separating the real and imaginary parts and defining the fluid velocity as \(\vec{v}=\
abla S/m\), it is possible to rewrite the GPP system as [22; 23]:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho\vec{v}) = 0, \tag{4}\] \[\frac{\partial(\rho\vec{v})}{\partial t}+\
abla\cdot(\rho\vec{v }\otimes\vec{v}+p_{\mathrm{SI}}I) = -\rho\
abla(Q+V),\] (5) \[\
abla^{2}V = 4\pi G\rho, \tag{6}\]  
where \(p_{\mathrm{SI}}=\frac{g}{2}\rho^{2}\) is the self-interaction pressure, and \(Q=-\frac{\hbar^{2}}{2m}\frac{\
abla^{2}\sqrt{\rho}}{\sqrt{\rho}}\) is the quantum potential. In this framework, the GPP system is known as the quantum hydrodynamic formulation. If we take the limit \(\hbar/m\to 0\), the quantum potential becomes zero, and the classical hydrodynamic equations are recovered with a polytropic equation of state with polytropic constant \(K=g/2\) and polytropic index \(n=1\). In this limit, we can see the effect of the nonlinear term in the GP equation classically, which results from two-body dispersion interactions between bosons:* Repulsive self-interaction (\(g>0\)): the gas has positive pressure opposing the gravitational force, allowing stable structures. * Attractive self-interaction (\(g<0\)): no classical behavior is possible since negative pressure is not physically acceptable. * Without self-interaction (\(g=0\)): the limit of positive self-interaction when the polytropic constant tends to zero corresponds to a dustlike state. When \(h/m>0\), the quantum potential reintroduces the possibility of structure formation regardless of the value of \(g\). The quantum potential \(Q\) is crucial for this behavior. As in any macroscopic system, global quantities can be measured. Some of these are:  
\[M_{T} = m\int|\Psi|^{2}d^{3}x=\int\rho d^{3}x, \tag{7}\] \[W = \frac{m}{2}\int\Psi^{*}V\Psi d^{3}x=\frac{1}{2}\int\rho Vd^{3}x,\] (8) \[K = -\frac{\hbar^{2}}{2m}\int\Psi^{*}\
abla^{2}\Psi d^{3}x\] (9) \[= \frac{1}{2}\int\rho|\vec{v}|^{2}d^{3}x+\frac{1}{2}\int|\
abla \sqrt{\rho}|^{2}d^{3}x\] \[\
abla^{2}V=4\pi Gm|\Psi|^{2}. \tag{2}\]0 \[\
abla^{2}V=4\pi Gm|\Psi|^{2}. \tag{2}\]1  
where \(M_{T}\) is the mass, \(W\) is the potential energy, \(K\) is the total kinetic energy (with \(K_{v}\) from classical contributions and \(K_{\rho}\) from quantum effects), and \(I\) is the self-interaction energy.