  A theoretical model of a Bose-Einstein condensate with angular spin-orbit
coupling has recently been proposed and it has been established that a
half-skyrmion represents the ground state in a certain regime of spin-orbit
coupling and interaction. Here we investigate low-lying excitations of this
phase by using the Bogoliubov method and numerical simulations of the
time-dependent Gross-Pitaevskii equation. We find that a sudden shift of the
trap bottom results in a complex two-dimensional motion of the system's center
of mass that is markedly different from the response of a competing phase, and
comprises two dominant frequencies. Moreover, the breathing mode frequency of
the half-skyrmion is set by both the spin-orbit coupling and the interaction
strength, while in the competing state it takes a universal value. Effects of
interactions are especially pronounced at the transition between the two
phases.
