  Recently, skyrmions with integer topological charges have been observed
numerically but have not yet been shown rigorously on two-component systems of
nonlinear Schrodinger equations (NLSE) describing a binary mixture of
Bose-Einstein condensates. Besides, half-skyrmions characterized by
half-integer topological charges can also be found in the nonlinear sigma model
which is a model of the Bose-Einstein condensate of the Schwinger bosons. Here
we prove rigorously the existence of half-skyrmions which may come from a new
type of soliton solutions called spike-vortex solutions of two-component
systems of NLSE on the entire plane. These spike-vortex solutions having spikes
in one component and a vortex in the other component may form half-skyrmions.
By Liapunov-Schmidt reduction process, we may find spike-vortex solutions of
two-component systems of NLSE.
