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Recent studies have demonstrated that defocusing cubic nonlinearity withlocal strength growing from the center to the periphery faster than $r^{D}$, inspace of dimension $D$ with radial coordinate $r$, supports a vast variety ofrobust bright solitons. In the framework of the same model, but with a weakerspatial-growth rate, $\sim r^{\alpha }$ with $\alpha \leq D$, we here test thepossibility to create stable\textit{\ localized continuous waves} (LCWs) inone- and two-dimensional (1D and 2D) geometries, \textit{% localized darksolitons} (LDSs) in 1D, and \textit{localized dark vortices} (LDVs) in 2D,which are all realized as loosely confined states with a divergent norm.Asymptotic tails of the solutions, which determine the divergence of the norm,are constructed in a universal analytical form by means of the Thomas-Fermiapproximation (TFA). Global approximations for the LCWs, LDSs, and LDVs areconstructed on the basis of interpolations between analytical approximationsavailable far from (TFA) and close to the center. In particular, theinterpolations for the 1D LDS, as well as for the 2D LDVs, are based on a\textquotedblleft deformed-tanh" expression, which is suggested by the usual 1Ddark-soliton solution. In addition to the 1D fundamental LDSs with the singlenotch, and 2D vortices with $S=1$, higher-order LDSs with multiple notches arefound too, as well as double LDVs, with $S=2$. Stability regions for the modesunder the consideration are identified by means of systematic simulations, theLCWs being completely stable in 1D and 2D, as they are ground states in thecorresponding settings. Basic evolution scenarios are identified for thosevortices which are unstable. The settings considered in this work may beimplemented in nonlinear optics and in Bose-Einstein condensates.

Localized dark solitons and vortices in defocusing media with spatially inhomogeneous nonlinearity