  We report comprehensive Monte-Carlo studies of the melting of skyrmion
lattices in systems of small, medium, and large sizes with the number of
skyrmions ranging from $10^{3}$ to over $10^{5}$. Large systems exhibit
hysteresis similar to that observed in real experiments on the melting of
skyrmion lattices. For sufficiently small systems which achieve thermal
equilibrium, a fully reversible sharp solid-liquid transition on temperature
with no intermediate hexatic phase is observed. A similar behavior is found on
changing the magnetic field that provides the control of pressure in the
skyrmion lattice. We find that on heating the melting transition occurs via a
formation of grains with different orientations of hexagonal axes. On cooling,
the fluctuating grains coalesce into larger clusters until a uniform
orientation of hexagonal axes is slowly established. The observed scenario is
caused by collective effects involving defects and is more complex than a
simple picture of a transition driven by the unbinding and annihilation of
dislocation and disclination pairs.
