  Motivated by recent experiments on the supersolid behavior of $^4$He, we
study the effect of an edge dislocation in promoting superfluidity in a Bose
crystal. Using Landau theory, we couple the elastic strain field of the
dislocation to the superfluid density, and use a linear analysis to show that
superfluidity nucleates on the dislocation before occurring in the bulk of the
solid. Moving beyond the linear analysis, we develop a systematic perturbation
theory in the weakly nonlinear regime, and use this method to integrate out
transverse degrees of freedom and derive a one-dimensional Landau equation for
the superfluid order parameter. We then extend our analysis to a network of
dislocation lines, and derive an XY model for the dislocation network by
integrating over fluctuations in the order parameter. Our results show that the
ordering temperature for the network has a sensitive dependence on the
dislocation density, consistent with numerous experiments that find a clear
connection between the sample quality and the supersolid response.
