  We consider dark solitons and their stability in spin-orbit coupled polariton
condensates. The system supports spinor solitons of two types: conventional
(symmetric) dark solitons and asymmetric half-dark solitons. They demonstrate
essentially different behavior upon variation of the strength of spin-orbit
coupling. One-dimensional spin-orbit coupled dark solitons are usually
unstable, while half-dark solitons can be stable. Two-dimensional dark solitons
at early stages of the development of transverse instabilities turn into
asymmetric snaking patterns and later into sets of vortex-antivortex solitons
with notably different shapes. Depending on the sign of spin-orbit coupling two
distinct instability scenarios are possible for such solitons, in which
vortices in one component correspond to vortices or antivortices in other
component. The decay of two-dimensional half-dark solitons results in the
formation of half-vortex chains.
