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We study the dynamics of bright and dark matter-wave solitons in the presenceof a spatially varying nonlinearity. When the spatial variation does notinvolve zero crossings, a transformation is used to bring the problem to astandard nonlinear Schrodinger form, but with two additional terms: aneffective potential one and a non-potential term. We illustrate how to applyperturbation theory of dark and bright solitons to the transformed equations.We develop the general case, but primarily focus on the non-standard specialcase whereby the potential term vanishes, for an inverse square spatialdependence of the nonlinearity. In both cases of repulsive and attractiveinteractions, appropriate versions of the soliton perturbation theory are shownto accurately describe the soliton dynamics.

Matter-Wave Solitons in the Presence of Collisional Inhomogeneities: Perturbation theory and the impact of derivative terms