  Vortex cores in a superconductor can develop structure and manifest competing
orders. In strong magnetic fields, the inter-vortex distance can become short
enough for vortex cores to overlap, giving rise to long ranged textures. We
show that spin orbit coupling (SOC) can modify the interaction between vortex
cores to give rise to undulating competing order fields. We illustrate this
using the SO(3) theory of competing orders, wherein superconductivity and a
scalar order form a three-component vector field. We consider a spherical
surface with a radial magnetic flux that creates two vortices. We find
stationary solutions where both vortex cores develop competing order in (a) the
same sense, and (b) opposite senses. The latter represents a topologically
stable vector configuration analogous to a single skyrmion. Its free energy is
lowered when SOC is introduced, making it the ground state beyond a threshold
SOC strength. We study this physics on the two dimensional plane using the
attractive Hubbard model with Rashba SOC. Here, charge density wave (CDW) order
competes with superconductivity. We find phases with long ranged, but spatially
modulated, CDW order where the average CDW moment vanishes. In a wide range of
parameters, CDW loses its rigidity as SOC lowers the energy cost for domain
wall formation. We discuss consequences for systems such as the cuprates where
charge order develops without a sharp diffraction peak.
