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In this paper, we develop a Newtonian approach to the collapse of galaxy fluctuations of scalar field dark matter under initial conditions inferred from simple assumptions. The full relativistic system, the so-called Einstein-Klein-Gordon system, is reduced to the Schr\"odinger-Newton one in the weak-field limit. The scaling symmetries of the SN equations are exploited to track the nonlinear collapse of single scalar matter fluctuations. The results can be applied to both real and complex scalar fields.
Guzmán et al. (Fri,) studied this question.