  In the present paper, using a molecular dynamics simulation, we study a
nature of melting of a two-dimensional ($2D$) system of classical particles
interacting through a purely repulsive isotropic core-softened potential which
is used for the qualitative description of the anomalous behavior of water and
some other liquids. We show that the melting scenario drastically depends on
the potential softness and changes with increasing the width of the smooth
repulsive shoulder. While at small width of the repulsive shoulder the melting
transition exhibits what appears to be weakly first-order behavior, at larger
values of the width a reentrant-melting transition occurs upon compression for
not too high pressures, and in the low density part of the $2D$ phase diagram
melting is a continuous two-stage transition, with an intermediate hexatic
phase in accordance with the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY)
scenario. On the other hand, at high density part of the phase diagram one
first-order transition takes place. These results may be useful for the
qualitative understanding the behavior of water confined between two
hydrophobic plates.
