## 1 Introduction  
The study of solitonic cores in physical systems has attracted significant interest due to their relevance in various fields, including cosmology, astrophysics, and condensed matter physics. These localized, stable structures arise from the balance between attractive and repulsive forces, leading to unique properties and behaviors.  
One specific field where this system has become particularly interesting is that of bosonic dark matter. This model assumes that dark matter particle is a spinless ultralight boson with a mass of order \(10^{-19}\)-\(10^{-23}\) eV/\(c^{2}\), and there are various reviews that draw a general panorama of the subject [1; 2; 3; 4; 5]. The dynamics of this type of matter in the mean-field approximation is ruled by the Gross-Pitaevskii-Poisson (GPP) system of equations, and the system can be seen as a Bose gas trapped by the gravitational field sourced by itself [6]. When the gas has no self-interaction, it is called the Fuzzy Dark Matter (FDM) regime [7] and is the main stream of the subject that has led to study both local (e.g., [8; 9]) and structure formation dynamics [10; 11; 12].  
The interesting thing about this model is that for this ultralight particle, the de Broglie wavelength is large, and thus the structures have a minimum size. In fact, it was found since the breakthrough simulation in [10] that cores were an essential part of structures and were an attractor profile surrounded by an envelope with high kinetic energy. This meant that these cores could be fitted by an empirical density profile that coincided numerically with the solution of the ground-state solutions of the GPP system [13]. Ever since, these cores have been considered literally the keystone of structures in FDM. Bounds on the values of self-interactions arise from local and cosmological constraints (see, e.g., [14; 15; 16; 17; 18]).  
The generalization to include self-interacting bosons is a natural extension, and limits to the self-interaction regime are a matter of interest, because the dark matter distribution differs from that of FDM [19], time scales of saturation and relaxation change [20], and ultimately, there is a Tomas-Fermi regime with rather simple density profiles [21].  
This background leads us to revisit some properties and stability of solitonic cores with self-interaction that prove to be useful. We solve the well-known eigenvalue problem for ground-state solutions of the GPP system, with a rather unusual but efficient method that involves genetic algorithms. We then propose an empirical formula that describes its profile. We then study the stability of the solutions and those associated with the empirical formula. We compare the reaction of both to truncation error perturbations in both amplitude and frequencies of the oscillation modes triggered. We point out interesting differences and potential implications within the dark-matter context.  
This paper is organized as follows: Section 2 provides an overview of the theoretical framework and equations governing solitonic cores. Section 3 presents the methodology used to construct the ground-state solutions. Section 4 contains a comparison with empirical formulas, while in Section 5, we compare the evolution of solutions. Finally, Section 6 draws some conclusions.