  We find Skyrmion-like topological excitations for a two-dimensional spin-1/2
system. Expressing the spinor wavefunction in terms of a rotation operator maps
the spin-1/2 system to a Manakov system. We employ both analytical and
numerical methods to solve the resulting Manakov system. Using a generalized
similarity transformation, we reduce the two-dimensional Manakov system to the
integrable one-dimensional Manakov system. Solutions obtained in this manner
diverge at the origin. We employ a power series method to obtain an infinite
family of localized and nondiverging solutions characterized by a finite number
of nodes. A numerical method is then used to obtain a family of localized
oscillatory solutions with an infinite number of nodes corresponding to a
skyrmion composed of concentric rings with intensities alternating between the
two components of the spinor. We investigate the stability of the skyrmion
solutions found here by calculating their energy functional in terms of their
effective size. It turns out that indeed the skyrmion is most stable when the
phase difference between the concentric rings is $\pi$, i.e., alternating
between spin up and spin down. Our results are also applicable to doubly
polarized optical pulses.
