  We study statically homogeneous Bose-Einstein condensates with spatially
inhomogeneous interactions and outline an experimental realization of
compensating linear and nonlinear potentials that can yield constant-density
solutions. We illustrate how the presence of a step in the nonlinearity
coefficient can only be revealed dynamically and consider, in particular, how
to reveal it by exploiting the inhomogeneity of the sound speed with a
defect-dragging experiment. We conduct computational experiments and observe
the spontaneous emergence of dark solitary waves. We use effective-potential
theory to perform a detailed analytical investigation of the existence and
stability of solitary waves in this setting, and we corroborate these results
computationally using a Bogoliubov-de Gennes linear stability analysis. We find
that dark solitary waves are unstable for all step widths, whereas bright
solitary waves can become stable through a symmetry-breaking bifurcation as one
varies the step width. Using phase-plane analysis, we illustrate the scenarios
that permit this bifurcation and explore the dynamical outcomes of the
interaction between the solitary wave and the step.
