  We study the dynamics of bright and dark matter-wave solitons in the presence
of a spatially varying nonlinearity. When the spatial variation does not
involve zero crossings, a transformation is used to bring the problem to a
standard nonlinear Schrodinger form, but with two additional terms: an
effective potential one and a non-potential term. We illustrate how to apply
perturbation theory of dark and bright solitons to the transformed equations.
We develop the general case, but primarily focus on the non-standard special
case whereby the potential term vanishes, for an inverse square spatial
dependence of the nonlinearity. In both cases of repulsive and attractive
interactions, appropriate versions of the soliton perturbation theory are shown
to accurately describe the soliton dynamics.

Quantum Physics