  Vortex lattices, arising from repulsive inter-vortex interactions, are a
canonical example of emergent phenomena. Recent studies on the cuprates have
drawn attention to the appearance of competing correlations within vortex
cores. This gives internal structure to each vortex and changes the nature of
inter-vortex interactions, especially at short range. This can have significant
consequences for vortex lattices wherein vortices are in close proximity to one
another. We consider the attractive Hubbard model, a prototypical model for
phase competition, with checkerboard charge density wave order appearing in
vortex cores. We use next-nearest-neighbour hopping, $t'$, to tune the strength
of competing order in the core. Using Bogoliubov deGennes mean field
simulations, we study the variation in quasiparticle gap, superfluid stiffness
and shear stiffness, as the core order is tuned. The presence of competing
order allows for a new defect: domain walls that separate regions with
different types of charge order overlying the vortex-laden superfluid
background. We find the energy cost of such domain walls. Our results indicate
a stable intermediate supersolid region with coexisting superfluid and charge
orders. At one end, this phase is unstable to melting of the vortex lattice
into a purely charge ordered state, possibly via an intervening liquid-like
state. At the other end, charge order becomes incoherent due to domain
formation while superconducting order persists.
