We explore the consequences of including the repulsive three-particle interaction in the model of Bose-Einstein condensate dark matter model or fuzzy dark matter. Such a model based on properly modified Gross-Pitaevskii equation is intended to describe the distribution of dark matter particles in the highly dense regions, which correspond to the galaxy core and/or to the overlap of colliding galaxies. Specifically, we deal with the ${\varphi }^6$-model in terms of the macroscopic wave function of the condensate, where a locality of interaction is guaranteed by a large correlation length assumed to hold. After calculation of main thermodynamical characteristics, we find strong evidence of the existence of two distinct phases of dark matter, within its core, separated by the instability region lying between two differing special values of the pressure acting in the model. Some implications stemming from the existence of two phases and the related first-order phase transition are discussed.

We establish the relation of the second virial coefficient of a recently proposed $(\tilde{\mu },q)$-deformed Bose gas model [A. M. Gavrilik and Yu. A. Mishchenko, Ukr. J. Phys. 58, 1171 (2013)] to the interaction and compositeness parameters when either of these factors is taken into account separately. When the interaction is dealt with, the deformation parameter becomes linked directly to the scattering length and the effective radius of interaction (in general, to scattering phases). The additionally arising temperature dependence is a feature absent in the deformed Bose gas model within the adopted interpretation of the deformation parameters $\tilde{\mu }$ and $q$. Here the problem of the temperature dependence is analyzed in detail and its possible solution is proposed.

We elaborate further the м-deformation-based approach to the modeling of dark matter, in addition to the earlier proposed use of м-deformed thermodynamics. Herein, we construct м-deformed analogs of the Lane–Emden equation (for density profiles) and find their solutions. Using these, we plot the rotation curves for a number of galaxies. Different curves describing the chosen galaxies are labeled by respective (different) values of the deformation parameter м. As a result, the use of м-deformation leads to the improved agreement with observational data. For all the considered galaxies, the obtained rotation curves (labeled by м) agree better with data, as compared to the well-known Bose–Einstein condensate model results of T. Harko. Besides, for five of the eight cases of galaxies, we find a better picture for rotation curves, than the corresponding Navarro–Frenk–White (NFW) curves. The possible physical meaning of the parameter м basic for this version of м-deformation is briefly discussed.

A new version of the q-deformed supersymmetric quantum mechanics (q-SQM), which is inspired by the Tamm–Dankoff-type (TD-type) deformation of quantum harmonic oscillator, is constructed. The obtained algebra of q-SQM is similar to that in Spiridonov’s approach. However, within our version of q-SQM, the ground state found explicitly in the special case of superpotential yielding q-superoscillator turns out to be non-Gaussian and takes the form of special (TD-type) q-deformed Gaussian.