Matter-Wave Solitons in the Presence of Collisional Inhomogeneities:   Perturbation theory and the impact of derivative terms
  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. 
S. Middelkamp, P.G. Kevrekidis, D.J. Frantzeskakis, P. Schmelcher