Ultralight axion is a dark matter candidate with mass and de Broglie wavelength of order kiloparsec. Such an axion, also called fuzzy dark matter (FDM), thermalizes via gravitational force and forms a Bose–Einstein condensate. Recent studies suggested that the quantum pressure from FDM can significantly affect structure formation in small scales, thus alleviating the so-called "small-scale crisis." In this paper, we develop a new technique to discretize the quantum pressure and illustrate the interactions among FDM particles in an N-body simulation that accurately simulates the formation of the dark matter halo and its inner structure in the region outside the softening length. In a self-gravitationally bound virialized halo, we find a constant density solitonic core, which is consistent with theoretical prediction. The existence of the solitonic core reveals the nonlinear effect of quantum pressure and impacts structure formation in the FDM model.
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