Ultra-Light Axion Dark Matter and its impacts on dark halo structure in
  $N$-body simulation
  The Ultra-Light Axion (ULA) is a dark matter candidate with mass
$\mathcal{O}(10^{-22})$eV and de Broglie wavelength of order kpc. Such an
axion, also called the Fuzzy Dark Matter (FDM), thermalizes via the
gravitational force and forms a Bose-Einstein condensate. Recent studies
suggested that the quantum pressure from the FDM can significantly affect the
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 the
$N$-body simulation, which 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 the theoretical prediction.
The existence of the solitonic core reveals the non-linear effect of quantum
pressure and impacts the structure formation in the FDM model.
